# Geometric Measurement Formulas - A Teacher's Perspective

This module provides a systematic and coherent development of the measurement formulas for computing distances, area and volume of geometric figures in the plane and in space including all of the formulas discussed in high school geometry. It also discusses a variety of problems and results in plane and solid geometry that relate to distance, area and volume as well as the rich history of geometric measurements.

The required textbook for this module is Mathematics For High School Teachers by Usiskin, Peressini, Marchisotto and Stanley. The corresponding content for this module is drawn from Chapters 8 and 10 of the text and the module is written so that it can be completed without reference to other chapters of the book.

This module is divided into the following units:

• Unit 1: Measuring distance in ordinary and unusual ways.
• Unit 2: Understanding the area formulas of plane geometry.
• Unit 3: Understanding the volume and surface area formulas of solid geometry.

These units deal with topics that are either in the typical high school geometry curriculum or are closely related to topics taught high school geometry. However, these topics are treated from a more advanced standpoint that assumes some of the collegiate background that prospective high school mathematics teachers take as undergraduates. This module complements the textbook treatment of this content with web-based resources and dynamic geometric sketches that are not available in any print textbook.

Common Core Standards for Mathemtical Practice that are emphasized include:

• 1. Make sense of problems and persevere in solving them.
• 3. Construct viable arguments.
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.

The author of this module is Tony Peressini, Professor Emeritus of Mathematics at the University of Illinois. This module was completed in March 2004

# Detailed Description

Module Objectives:
Geometric measurement formulas; that is, formulas for computing:

• the area or perimeter of geometric figures in the plane such as triangles, polygons and circles,
• the volume or surface area of various geometric solids such as pyramids, cylinders and spheres, and
• the distance between geometric objects such as the distance between two points or between a point and a line,

are important topic in high school geometry.

The discussion of basic area and perimeter formulas usually begins in middle school or even earlier when students begin to identify and classify geometric figures. From that point on, geometric measurement formulas are important tools in all mathematics courses.

The first objective of this module is to provide mathematics teachers with a systematic development of all of the standard geometric measurement formulas with careful attention to their rich history and their numerous applications. The second objective is to help teachers to explore the many connections that exist between these formulas and the rest of geometry, algebra, precalculus and calculus.

We will help teachers achieve these instructional objectives through the guided individual study of text and internet-based course resources, four required problem assignments and the preparation of an approved final classroom project.

Textbook and prerequisite skills for this module.

The required textbook for this module is Mathematics for High School Teachers by Usiskin, Peressini, Marchisotto and Stanley. This book is a very good professional development resource for high school mathematics teachers. If you do not wish to purchase it yourself, it may be available in your school library or you may ask your school librarian to order a copy.

All four assignments for this module will require you to produce accurate geometric sketches and constructions and insert them into the Microsoft Word documents that you will prepare for the four assignments of this module. Consequently, you will need access to and some facility with a geometric construction program such as Geometer's Sketchpad to do the assignments. Math Teacher Link offers a Tutorial for math teachers (Module 4: Using the Geometer's Sketchpad) that you can take for free as an MTL Guest or for credit (SeeRegistration , Fees and Getting Started for more details.)

You will submit these assignments and retrieve your graded assignments electronically through the MTL Module Working Environment (also referred to as Moodle)..

Structure of the Module:

This module is divided into the following units and parts:

Unit 1: Measuring Distance in Ordinary and Unusual Ways

• Part 1: What is distance?
• Part 2: Minimum distance problems in plane geometry.

Unit 2: Understanding the Area Formulas of Plane Geometry.

• Part 1: Area formulas for polygonal figures
• Part 2: Area formula for curvilinear figures.

Unit 3: Understanding the Volume and Surface Area formulas of Solid Geometry

• Part 1:Volume formulas for Polyhedra.
• Part 2: Volume and Surface Area formulas for cylinders, spheres, cones and other "curvy" solids.

Module Completion Requirements:

You wil have an enrollment period of three months from the date that you are e-mailed your MTL log-in, password and 800-Help number to complete all of your work for this module. A three month extension is possible for unforeseen circumstances provided that you are making reasonable progress with your work.

There are a total of four assignments: Assignment 1.1 for Unit 1, Assignments 2.1 and 2.2 for Unit 2, and Assignment 3.1 for Unit 3. You should do each assignment as soon as you complete the parts of the unit required by the assignment. You will be asked to place the solutions of all of the problems in a given assignment into a single Microsoft Word document and then submit the document electronically through the MTL Module Working Environment (also referred to as the ClassComm system). The module instructor will evaluate your work on each assignment and return your graded assignment through ClassComm and /or e-mail the results to you.

Teachers enrolled in this module for University of Illinois Continuing Education Units are required to satisfactorily complete these four assignments.

Teachers enrolled for University of Illinois graduate credit, or undergraduate prospective mathematics teachers enrolled for University of Illinois undergraduate credit, must also complete an approved Final Project.

# Required Materials

In addition to the general requirements for participating in Math Teacher Link, this module also has the following additional materials requirement:

• The Geometer's Sketchpad application (version 4 or higher) needs to be installed on the computer you will use.
Go to the Geometer's Sketchpad home page of Key Curriculum Press, the makers of the software. This page has information, resources and support for The Geometer's Sketchpad software. Follow their information and pricing link to order the current version of Geometer's Sketchpad software for either your Mac or Windows machine (Approximately \$70.00 - single user). Install the program on your machine according to the instructions that come with the program.
• We have stayed with Geometer's Sketchpad rather than go with GeoGebra because we find that using well supported and well documented software saves valuable time and aligns well with other published materials.  It is the same reason we use Word rather than OpenOffice for our word processing.
• You need to obtain a copy of Mathematics For High School Teachers by Usiskin, Peressini, Marchisotto and Stanley.

This book is a very good professional development resource for high school mathematics teachers. If you do not wish to purchase it yourself, it may be available in your school library or you may ask your school librarian to order a copy. You should proceed with arrangements to obtain a copy of the book as soon as you submit your registration for this module so that you can begin your work on the module without delay when you receive your MTL Log-in and password.

# Step by Step Instructions

Enrollment and Credit Options: Mathematics teachers who have completed an undergraduate major in mathematics or mathematics education or the equivalent can enroll in this module for either one-quarter unit of University of Illinois Graduate Credit. Illinois teachers also have the option of enrolling for three Continuing Education Units, CEU's are applicable to teacher recertification in Illinois. Within 5 working days of completion of enrollment, these teachers will be issued a log-in and password that will provide them access to the Module Working Environment (also referred to as ClassComm) where they submit required assignments and retrieve the graded results.

Mathematics teachers may also enroll as MTL Guests (free). Guest registration gives teachers access to the instructional materials but no access to instructional support. including homework submission and grading,or the e-mail help systems.

Teachers enrolled for graduate credit must complete and submit the four assignments for the module described in Steps #1 through #7 below and an approved Final Project (See Step #8 below for more details about this requirement.)

Teachers enrolled for Continuing Educations Units are not required to complete Final Project but they are required to complete and submit the assignments for the module described in Steps #1 through #7 below. Continuing Education Units are not recorded on a University transcript; However, the University of Illinois Division of Academic Outreach provides a Letter of Completion for a teacher who completes a module for continuing education units.

Mathematics teachers may also enroll as MTL Guests (free). Guest registration gives them access to instructional materials but no access to instructional support, or to the Module Working Environment (also referred to as ClassComm).

Estimated time requirements for completion of the module: We estimate that the work for this module will take participating teachers an average of about 45 hours to complete. Technically, you have a 3-month enrollment period to complete the module after you enrollment is complete and you have been sent your log-in and password and 800-number Help information. However, we recommend that you have as a goal to complete the module in eight weeks or less. That should be possible even when you are teaching if you set aside three or four hours each week to work on the module.

#### After you have registered for Module 15 proceed as follows: Obtain a copy of the textbook Mathematics For High School Teachers by Usiskin, Peressini, Marchisotto and Stanley. Although you will be using only two chapters of this book for this module, this book is an excellent resource for your professional library or for your school library. Although there is a place on the publishers web site for requesting an examination copy, such copies are sent only to college instructors of courses for which the text may be adopted.

After you have obtained a copy of the textbook, proceed to complete the following steps:

# Step #1

STEP #1: Complete Unit 1, Part 1: What is distance?. Complete Part 1 0f Assignment 1.1 according to the directions at the beginning of that assignment. Save that document to combine it with Part 2 of Assignment 1.1 in Step #2 below.

Comments on Step #1: This part of the unit explores the concept of distance in several different contexts, some very familiar and some very unusual such as taxicab distance in the plane and the Hamming distance, which is very useful in codes and data transmission.

Create a Module 15 Assignments folder on your hard drive and place a copy of your completed Assinment 1.1 Part 1 in that folder. Then submit Assignment 1.1 Part 1 by logging in to the Module Working Environment (also referred to as Moodle) for Module 15 using the Hand In tab at the top of this page and the login directions that were in your welcome letter.

# Part 1: What is distance?

### Companion Content For Section 8.1.1 of Mathematics for High School Teachers.

#### Teacher's Perspective

For most high school students, the mathematical term "distance" has a very precise and accepted meaning: It applies to two points P and Q on the number line or in a coordinate plane or in coordinate 3-dimensional space, and it is measured by one of the following formulas:

These formulas describe the so-called Euclidean distance between the points P and Q. But, as you will soon see, Euclidean distance is not the only way in which the concept of distance is used and is useful in mathematical contexts.

The purpose of Section 8.1.1 of your Mathematics For High School Teachers text is to illustrate a few of the many other ways in which the concept of distance is used and also to arrive at an axiomatic definition of a distance function or metric.

As you study this section and do the problems in the assignment, think of other ways in which you have seen the idea of distance used in mathematical and non-mathematical contexts and ask yourself if these instances are examples of distance functions in the sense of the Definition at the end of Section 8.1.1.

Read Section 8.1.1. Be sure to answer Questions 1 – 5 and check your responses with the answers at the end of the section.

# Just Do It!

### 1.1.1 Just Do It!

Suppose that the distance between two points P = (x, y) to Q = (u, v) in a rectangular coordinate plane is defined by d(P, Q) = the largest of the two numbers | x – u| and | y – v |; that is, d(P(x, y), Q(u, v)) = max{ | x – u| , y – v | }.

1. Find two points P and Q for which d(P, Q) is not equal to the Euclidean distance between P and Q, and two other points P and Q for which d(P, Q) is equal to the Euclidean distance between P and Q.
2. Given that the circle C(P, R) of radius R centered at a point P is defined to be the set of all points Q in the plane such that Q is at a distance R from P. describe geometrically the circle C(P(0, 0) , 1) of radius 1 centered at the origin (0, 0)..
3. Does this definition of a distance on the plane satisfy the conditions of a distance function?

1. Most choices of two points P and Q will result in different values for d(P, Q) and the Euclidean distance between P and Q. For example, if P = (3, 4) and Q = (0, 0), then the Euclidean distance between P and Q is 5 but d(P, Q) = 4. On the other hand, for the same choice of Q and for P = (0, 1), (1,0), (- 1, 0), (0, -1), d(P, Q) and the Euclidean distance are both equal to 1.
2. The circle C(P(0, 0) , 1) for d(P, Q) is actually the square centered at (0, 0) with vertices at (1, 1), (1, - 1), (- 1, - 1), (1, - 1):

For example, for any point Q = (x, y) on the left side of this square, x = - 1 and y is between - 1 and + 1,
so d(P, Q) = max{| -1|, | y |} = 1.
3. For any two points, P = (x, y) and Q = (u, v), it is true that:
D1: d(P, Q ) is non-negative because the maximum of two non-negative numbers is non-negative.;
D2: d(P, Q ) = 0 if and only if P = Q because the maximum of the numbers | x – u | and
| y – v | is 0 if and only if x = u and y = v; that is, if and only if P = Q;
D3: d(P, Q) = d(Q, P) because |x – u | = | u – x | and | y – v | = | v – y | by properties of absolute value.
D4: For any third point R = (s, t), the definition of d and the triangle inequality for absolute value show that:

# Assignment 1.1, Part 1

Assignment Completion and Submission Directionss: Prepare a single MSWord document for both parts of Assignment 1.1. Use Equation Editor or Math Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create inport these diagrams or graphs from other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete both parts of Assignment 1.1, submit your Word file using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

Solve Problems 3, 5, 10 and 12 in 8.1.1 Problems at the end of this section 8.1.1 of your text. Your solutions should be placed in a MS Word document that will be the Part 1 of Assignment 1.1 for this module.

# Step #2

STEP #2: Complete Unit 1, Part 2: Minimum distance problems in plane geometry. Complete and submit Part 2 of Assignment 1.1 according to the directions at the beginning of that assignment.

Comments on Step #2: This part of the unit discusses a number of very interesting minimum distance problems in plane geometry including Hero's Problem, which has applications to light reflection, and the Fermat and Fagnano problems. The internet-based dynamic and interactive sketches add a "hands on" aspect that make the text discussion come to life.

Note: Facility with geometric construction software such as Geometers Sketchpad is essential for creating some of the sketchs required in this part of Assignment 1.1 as well as in the remaining assignments for this module. If you do not have this prerequisite skill, you can acquire it either completing Module 4: Using the Geometer's Sketchpad for graduate or continuing education credit, or by registering as a Guest (free!) for that module and completing it on your own. If you have used Geometer's Sketchpad in the past but currently do not use it frequently, we strongly recommend that you refresh your background by completing Module 4 as an MTL Guest.

Save a copy of your completed  Assignment 1.1 Part 2 to the Module 15 Assignments folder on your hard drive. Then submit Assignment 1.1 Part 2 by logging in to the Module Working Environment (also known as Moodle) using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

# Part 2: Minimum distance problems in plane geometry.

### Companion Content For Section 8.1.2 of Mathematics for High School Teachers.(Some connections are made to related material from Sections 8.2.3 and 8.2.5.)

#### Distance in geometry.

As we have seen in Part 1 of this unit, the concept of distance is used in different ways and can be applied to measure the “apartness” of objects of very different sorts. However, in high school geometry, the term distance almost always refers to Euclidean distance d(x, y) between two points x and y in a coordinate plane or in coordinate 3-dimensional space.

Euclidean distance plays a central role in the description of important geometric objects such as:

• circles in a plane with given centers and radii,
• the perpendicular bisector of a given line segment in a plane,
• geometric descriptions of the conics (i.e. parabolas, ellipses and hyperbolas) in a plane,
• spheres in 3-dimensional space with a given center and radius,
• planes perpendicular to a given line in space at a given point on that line.

Euclidean distance (or length) is also a frequent ingredient in describing special properties of geometric figures. The Pythagorean Theorem for right triangles and the Law of Cosines for arbitrary triangles are among the most familiar and useful of such properties for triangles. For cyclic quadrilaterals (that is, quadrilaterals that can be inscribed in a circle) Ptolemy’s Theorem states that the product of the lengths of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the lengths of the two pairs of opposite sides.

For circles, Theorem 8.15 in Section 8.2.3 of your text states that if A, B, C and D are any four points of a circle and if the line AB intersects the line CD at a point E, then |AE| . |BE| = |CE| . |DE|.

As is observed in Section 8.2.5, of your text, this result has the following more familiar consequences concerning circles as corollaries:

• Theorem 1: If two chords AB and CD of a circle intersect at a point E inside the circle, then the product of the lengths of the segments AE and EB of the chord AB is equal to the product of the lengths of the segments CE and ED of the chord CD.
• Theorem: 2: If two secants are drawn to a circle from a point E outside of the circle, then the products of the length of the secant and the length of the external segment is the same for both secants.
• Theorem 3: If a secant and tangent are drawn to a given circle from a point E outside of the circle, then the square of the of the length of the tangent segment is equal to the product of the secant and the length of its external segment.

These are often called the Power Theorems in high school geometry texts.

#### Click here for a dynamic sketch that makes the preceding results plausible.

(The Download Folder for this module also includes a dynamic Geometers Sketchpad file Power_Th.gsp that illustrates the Power Theorem in a clearer and more dramatic manner.)

#### Minimum distance problems.

There are many important problems in geometry, calculus and applied mathematics that involve minimizing the distance between geometric objects

The purpose of Section 8.1.2 of your text Mathematics For High School Teachers is to discuss a few of the many minimum distance problems and to show how these problems can be solved with geometric tools.

Theorems 8.1 – 8.3 are familiar, intuitive results that are frequently stated and applied in high school and lower division college mathematics texts without proof. We discuss them here because most mathematics teachers have never seen synthetic geometric proofs of these results even though they underlie many calculations and concepts involving Euclidean distance in the courses that they teach.

The remainder of the section is devoted to the discussion of three minimum distance problems, Hero’s Theorem, which is the geometric basis of for the physical law that underlies the reflection of light, as well as the Fermat and Fagnano Problems for triangles.

Hero's Problem: Given two points A and B on one side of a line L in a plane, find the path of minimum length between A and B that touches L

#### A dynamic sketch that explains Hero's Theorem.

(The Download Folder for this module also includes a dynamic Geometers Sketchpad file Hero_Th_Th.gsp that illustrates Hero's Theorem from another standpoint.)

#### Hero (or Heron) the Elder of Alexandria (c. 100 B.C.) Biography

Fermat's Problem: For a given acute-angled triangle ABC, locate the point P whose Euclidean distances from A, B and C have the smallest possible sum.

The point P that solves Fermat's Problem is often called the Fermat point of the triangle ABC. Some authors also call it the Torricelli point or the first isogonic point of the triangle ABC.

#### Pierre Fermat (1601 - 1665) Biography

Fagnano's Problem: In a given acute-angled triangle ABC, inscribe a triangle UVW whose perimeter is as small as possible.

#### A dynamic sketch that helps you to solve the Fagnano Problem.

(The Download Folder for this module includes a dynamic Geometers Sketchpad file Fagnano_Th.gsp that is an enhanced version of the preceding web-based sketch of Fagnano's Theorem.)

#### Giulio Carlo Toschi di Fagnano (1715 - 1797) Biography

The Fermat point of a triangle is an example of a triangle center. The ancient Greeks discovered and determined some of the properties of the following four triangle centers:

1. The incenter of a triangle ABC is the point of intersection of the bisectors of the interior angles of the triangle at the vertices A, B and C. The incenter is the center of the inscribed circle of the triangle.
2. The circumcenter of a triangle ABC is the point of intersection of the perpendicular bisectors of the three sides of the triangle.The circumcenter of the triangle is the center of the circle circumscribing the triangle ABC.
3. The orthocenter of a triangle ABC is the point of intersection of the altitudes of the triangle.
4. The centroid of a triangle ABC is the point of intersection of the medians of the triangle. The centroid can also be described as the "balancing point" or center of gravity of the triangle if the triangle ABC is regarded as a thin plate of uniform density.

It is a remarkable fact that the circumcenter, orthocenter and centroid always line on the same line called the Euler line of the triangle. Walter Fendt's Special Lines and Circles in a Triangle is an excellent resource for exploring these centers, lines and circles dynamically.

Professor Clark Kimberling of the University of Evansville maintains a wonderful and very interesting web resource on Triangle Centers.

Theorem 8.1 is often take for granted in high school and college textbooks. The synthetic geometric proof given in the book may seem too difficult and one might expect that a proof using analytic geometry or calculus would be much easier.1. 2.1 Just Do It! asks you to carry out a such a proof for the sake of comparison.

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# Assignment 1.1, Part 2

Assignment Completion and Submission Directionss: Prepare a single MSWord document for both parts of Assignment 1.1. Use Equation Editor or Math Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create inport these diagrams or graphs from other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete Assignment 1.1, submit your Word file by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

Solve Problems 5, 6, 9, 17 in Section 8.1.2.

Your solutions should be places in a MS Word document that will be the Part 2 of Assignment 1.1 for Unit 1 of this module.

# Just Do It!

### 1.2.1 Just Do It!

a) Use analytic geometry or calculus to prove that the distance from the origin (0,0) to points (x, y) on the line L with equation
ax + by = c where c is non-zero and either a or b is non-zero is minimized at the point (x, y) where;

b) Explain why the line segment from the origin to the point on L computed in a) is perpendicular to L

.
c) Explain how the conclusions of parts a) and b} can be used to give another proof of Theorem 8.1.

a) To find the point (x, y) on the line L that minimizes the distance from the origin (0,0) to points (x, y) on the line L with equation ax + by = c, it would suffice to solve the following problem:

If b is non-zero, then y = (c – ax)/b and so we can rewrite this minimization problem in terms of x as:

Because f(x) is a quadratic function of x with a positive leading coefficient, it has its minimum at the vertex of its graph, and coordinates of the vertex are given by:

b) The slope of the line ax +by = c is - a/b if b is non-zero and the line is vertical if b is zero. The slope of the line segment S joining (0, 0) to the minimum point in a) is b/a if a is not zero, and it is vertical if a is zero. Thus, because either a or b must be non-zero, the line L is perpendicular to the line segment S in any case.

c) Given an arbitrary line L and an arbitrary point P not on L, translate P and L so that P is at the origin. Since translations preserve distance and perpendicularity, apply b) to conclude that the length of the line segment S joining P to L and perpendicular to L is the distance between P and L.

# Step #3

STEP #3: Complete Unit 2, Part 1: Area formulas for polygonal figures Complete and submit Part 1 and Part 2 of Assignment 2.1. Combine the two parts as a single document and submit it as Assignment 2.1 according to the directions at the beginning of Part 2 of that assignment

Comments on Step #3: Section 10.1.1 develops the axioms for planar area that are used to establish the area formulas for geometric figures in this and subsequent sections. The internet-based content compares the modern and classical approaches to the determination of the area of geometric formulas.

Sections 10.1.2 continues the development of area formulas for polygonal figures in the plane including Hero's formula and a variety of other formulas related to various triangle congruence tests in plane geometry.

Save copies of both parts of your completed  Assingment 2.1 to the Module 15 Assignments folder on your hard drive. Then submit Part 1 and Part 2 of  Assignment 2.1 by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

# Unit 2, Part 1: Area formulas for polygonal figures.

#### Why do many students have difficulty with area and volume calculations?

And many students do! Those of us who teach mathematics at the college level have often observed that many otherwise capable students enter college with a surprisingly weak grasp of the meaning and calculation of areas and volumes of geometric figures. Not only do they have difficulty recalling and applying area and volume formulas for standard figures but also they confuse area with volume or fail to recognize or assign appropriate dimensions in area or volume calculations. Interestingly enough, standardized test data also indicate that students attain a reasonably good grasp of area calculations in middle school but that this grasp actually weakens somewhat in later high school .assessments.

Improving this situation is not simply a matter of requiring students to memorize basic formulas for areas and volumes of standard figures. We believe that the problem is that students do not understand area and volume as planar and spatial measures and they fail to see the relationship between the area or volume of a geometric figure and the linear dimensions of that figure. For instance, a student who simply memorizes formulas may confuse the formulas for the area and the circumference of of a circle or the formulas for the volume and surface area of a sphere, but a student who understands these relationships will be able to choose the right one with a little reflection.

High school geometry books typically state many area and volume formulas and apply them to a variety of figures; however, few of the formulas are actually proved. While this may be reasonable for high school students, it is important that teachers have a deeper understanding of these formulas and their interdependence. Yet, college geometry courses for prospective teachers often do not provide a rigorous development of these measurement formulas Our objective in Units 2 and 3 of this module is to provide that sort of systematic development.

Part 1 of this unit will show how all of the formulas for areas of polygonal figures in the plane can be derived from the single formula for the area of a square that states that area of a square is the square of its side length. We will base these derivations on a set of axioms for the area function. Part 2 will extend and augment the area axioms to obtain formulas for the areas of circles and other regions bounded by line segments, circular arcs and other curves.

# Area and Volume Computations - A Historical Perspective.

When we are asked to compute the area or volume of a certain geometric figure, we usually use one of the following two approaches:

1) The formula approach: We use appropriate formula(s) for the area or volume of standard geometric figures that we have learned or that we can find in lists of such formulas. This may require breaking up the figure into several pieces and applying different formulas for each piece.
2) The calculus approach: We set up appropriate definite integrals that represent the required area or volume and then apply the tools of calculus (mainly, the Fundamental Theorem of Calculus) to compute the values of these integrals.

Both of these approaches have their origins in the geometry of the Ancient Greeks as it was codified if Euclid’s Elements (circa 300 B.C.). [See Section 7.1.1 of your text for a concise discussion of the Elements and the history surrounding his work.] There is also an outstanding web resource on Euclid’s Elements that includes sample propositions from all 13 books of Euclid’s Elements. The diagrams in the Elements are enabled with Java so that they are dynamic. This resource can be accessed at the following URL:

Although both the calculus and formula approaches to the calculation of area and volume had their origins in the Elements of Euclid, Euclid and his predecessors approach to area and volume calculations differed from both of these approaches in the following significant way. The Greek geometers of Euclid’s approached the question of finding the area and volume in the following significantly different way: Given a geometric figure F of unknown area or volume, they attempted to construct with compass and straightedge another geometric figure G with the same area or volume as F and for which the area or volume was known. That was it! No formula for the area or volume of F in terms of the linear dimensions of F was required. After all, if the area or volume of the constructed figure G can be computed from its linear dimensions, then the problem of determining the area or volume of the original figure F was solved as far as they were concerned.

19) For example, the Greeks knew that the area of a square is equal to the square of its side length. To compute the area of a parallelogram P or a circle C, they attempted to construct squares S(P) and S(C) of equal area to P and C by using compass and straightedge alone. They found the required construction of the square S(P) from P easily, but the construction of a square S(C) of equal area to a given circle C eluded their every effort and became the famous squaring the circle problem. Although the Greeks suspected that this construction was impossible, a mathematical proof of its impossibility was not given until the nineteenth century.

Solve the following "squaring a rectangle" problem (with a little help from your friends!).

# Just Do It!

Explain how you would “square a rectangle”; that is, given a rectangle with sides a and b, construct a square with area ab The following diagram may be helpful:

By the similarity of the two smaller right triangles in the diagram,

the length of the unknown side, which is the altitude of the largest triangle is c = the square root of ab. Therefore, the yellow square with sides c in the following diagram

“squares” the blue rectangle with sides a and b.

### There is a dynamic version of the preceding diagram in the module download folder.

Note: The length c of the side of the square that “squares” a rectangle with sides of length a and b is the geometric mean of a and b. The figure shown as a hint in this Just Do It! is a well known geometric construction for a line segment of length equal to the geometric mean of two line segments of length a and b. (See Theorem 10.11 in Section 10.1.5 in the text for this result )

# Historical Comments on Problem 13 in Assignment 2.1, Part 1

The content of this Problem 13 is essentially Proposition 43 of Book 1 in Euclid’s Element..It is paraphrased in modern language in Problem 13 as follows:

Proposition 43 in Book 1 of Euclid's Elements. Given a parallelogram ABCD and a point E on the diagonal AC, let the parallel to AD intersect AB at H and CD at I, and let the parallel to AB through E intersect AD at F and BC at G. Then the parallelograms HEGB and FDIE have equal areas.

Download Geometer's Sketchpad file: Prop_43. This file provides a dynamic sketch of Euclid's Proposition 43 that allows you to vary the point E along the diagonal AC or the parallelogram ABCD and compare the areas of the parallelograms FDIE and HEGB.

Hopefully, as you have seen in your work on Assignment 2.1, this result is relatively easy to prove, and, in fact, it is a reasonable exercise for capable high school geometry students. Nevertheless, as we will now explain, this result is a basic tool in Euclid’s approach to the calculation of areas of polygonal and even more general planar regions.

To understand the connection between the result in Problem 13 and area calculation, we first establish the following interesting results in Euclid’s Elements concerning parallelogram areas. .

Proposition 42 of Book 1: Given an angle D and given a triangle ABC, it is possible to construct a parallelogram with D as one vertex angle that has the same area as ABC.

Proof: Let E be the midpoint of BC. Construct an angle CEF equal to D at E where F is the point of intersection of the line parallel to BC through the point A. Let G be the point of intersection of the line through C parallel to EF.

The triangles ABE and AEC have the same area because they have the same height and have bases of equal length. Also, ECGF is a parallelogram whose area is twice that of the two triangles ABE and AEC since the two triangles and the parallelogram share a common height and base length. Therefore, the parallelogram ECGF has the same area as the given triangle ABC.

Download Geometer's Sketchpad file: Prop_42. This file provides a dynamic sketch of Euclid's Proposition 42 that allows you to vary the angle A and the triangle ABC and compare the areas of the parallelograms ECGF and the triangle ABC.

Euclid sharpened the preceding result to the following statement:

Proposition 44 of Book 1: Given an angle D, a line segment XY and a triangle ABC, it is possible to construct a parallelogram with D as one vertex angle and AB as one side that has the same area as the triangle ABC.

Proof: First of all, by Proposition 43 we can construct a parallelogram CEFG with the angle D at one vertex that has the same area as the triangle ABC. Then place the line segment XY and the parallelogram CEFG so that the side CG of the parallelogram and the line segment XY are on the same line with C coincident with Y as in the figure below.

On the left side of the figure, we show the given triangle ABC on top, the given angle in the the middle, and the given line segment XY at the bottom.

On the right side of the figure, the top figure shows the result of applying Proposition 43 to the given triangle ABC and given angle D to obtain a parallelogram CEFG with the same area as the triangle ABC and with the given angle D at the lower left corner E. The lower figure on the right shows the parallelogram CEFG and a segment XY placed so that XY and CG are on the same line with Y = C. The construction of the lower right figure is now completed as follows:

1. Draw the lines m and n parallel to the segment EC passing through X and G respectively . Draw the lines r and s parallel to the line segment CG passing through E andC. . The lines m and r intersect at the lower left corner P of the figure. The line through Pand C intersects n at the upper right corner Q of the figure.
2. Construct the line through Q that is parallel to CG. This line intersects m at the point W the line containing the segment EC at Z.

Download Geometer's Sketchpad file Prop_44d: This file provides a dynamic sketch of Euclid's Proposition 44 that allows you to vary the angle D, the triangle ABC and the line segment XY.

# Just Do It!

A student comes to you with the following qustion: I know that the formula for the area of a circle of radius R is either Pi R squared or 2 Pi R. Which is correct? Of course, you can give the student the correct answer but can you suggest a very simple way in which the student can identify the correct choice for himself?

You might explain it this way: Suppose that the radius R of the circle is measured in feet. Then what would the units for the quantity 2 Pi R? ( Answer: It would also be feet.). And what are the units for Pi R squared? (Answer: Square feet.) Well then, which is the correct formula for the area of a circle? (Answer: Must be Pi R squared because area is measured in square units. Thank you Mr. Smith.)

# Assignment 2.1 Part 1

Assignment Completion and Submission Directionss: Prepare a single MSWord document for both Part 1 and Part 2 of Assignment 2.1. Use Equation Editor or Mathe Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create these diagrams or graphs in other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete both parts of Assignment 2.1, submit your Word file by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

Solve Problems 3, 8, 10 and 13 in 10.1.1 Problems at the end of this section. Your solutions should be placed in a MS Word document that will be the Part 1 of Assignment 2.1 for this module.

# Assignment 2.1, Part 2

Assignment Completion and Submission Directionss: Prepare a single MSWord document for both Part 1 and Part 2 of Assignment 2.1. Use Equation Editor or Mathe Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create these diagrams or graphs in other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete both parts of Assignment 2.1, submit your Word file by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

Solve Problems 4 a, b, c, 6, 7, 9, 14 in 10.1.2. These problems are restated below, sometimes with helpful diagrams or hints, for your convenience. Your solutions should be places in a MS Word document that will be the Part 2 of Assignment 2.1 for this module

Problem 4.a. Use Hero’s Formula to find the area of a 3-4-5 triangle.
b. Find the area of a triangle with sides 13, 14 and 15.
c.Use the answers to part b to determine the lengths of the three altitudes of the triangle with sides 13, 14, and 15.

Note: Parts a and b show that the 3-4-5 and the 13-14-15 triangles have integer areas. Among all triangles whose side lengths are consecutive integers, those with integer area are rather sparse. For integers n < 200, the only (n-1)-n-(n+1) triangles with integer areas are n = 4, 14, 52, 194.

Extra Credit Problem: Write a computer program that verifies the preceding statement.

Problem 6. Use a geometric construction program to construct a dynamic version of Figure 13 for given triangle vertices A, B and C.(See static figure below.)

b. Identify all cyclic quadrilaterals shown in Figure 13 and explain why they are cyclic.

Problem 7. b. Prove that Brahmagupta’s Formula does not hold for all quadrilaterals.
c. Explain why Hero’s Formula is a special cae of Brahmagupta’s Formula.

Problem 9. Deduce the Law of Sines from Theorem 10.8.

# Step #4

STEP #4: Complete Unit 2, Part 2: Area formulas for curvilinear figures Complete and submit Assignment 2.2 according to the directions at the beginning of that assignment

Comments on Step #4: Archimedes applied the Method of Exhaustion to determine the exact area of a variety of geometric figures in the plane including circles and parabolic sections. Section 10.1.4 describes Archimedes' development of of the formula for the area and circumference of a circle and the determination of the value of Pi. The internet-based content explores the development of the Method of Exhaustion in calculus and also shows the very clever approach used by Archimedes to find the area of parabolic section. You will definitely learn some new and interesting things about parabolas here!

Save a copy of your completed  Assignment 2.2 to the Module 15 Assignments folder on your hard drive. Then submit Assignment 2.2 by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

# Part 2: Area formulas for curvilinear figures.

#### The Method of Exhaustion.

Most mathematics teachers encountered the Method of Exhaustion first when they studied the definite integral in calculus. However, the Method of Exhaustion predates and anticipates the development of calculus by many centuries, and the discussion of the computation of the area of a circle in Section 10.1.4 illustrates how the Method of Exhaustion was used long before the invention of calculus. In that discussion, the given circle C of radius r is inscribed with regular polygons s[n] and circumscribed with regular polygons S[n] with 2 to the power n sides in such a way that:

The area of the circle C is the unique positive real number that is larger than s[n] and smaller than S[n] for all positive integers n. Thus, the area of the circle is “exhausted” by the increasing areas of the inscribed regular polygons and the decreasing areas of the circumscribed regular polygons.

The application of the Method of Exhaustion to the definition and calculation of the definite integral in calculus is technically simpler than the preceding application to defining and computing the area of a circle as we will now see.

#### The Method of Exhaustion and the Definite Integral in calculus

One of the first applications of the definite integral that encountered in your calculus course is to the calculation of the area bounded by the graphs of two functions f and g over an interval [a, b] of the x-axis. For example, the graph below shows an area bounded by the graphs of two functions f and g over the interval [-3, 2] of the x-axis:

This area is computed with the following definite integral:

While this calculation is a routine application of calculus, it does use three of the most powerful tools in calculus: the definite integral, the anti-derivative, and The Fundamental Theorem Of Calculus. It is remarkable that Archimedes had developed the tools necessary to compute the areas of “parabolic sections” such as that given above long before calculus was invented. His tools were based on an ingenious application of the Method of Exhaustion and his discovery of a remarkable property of parabolas. As a high school math teacher, you may think that there is not much about parabolas that you have not seen before but I think that you will find the following information to be new and very interesting. At least I did!

# Archimedes and the Method of Exhaustion For Parabolic Sections

I am very grateful to my colleague, Professor Peter Braunfeld of the University of Illinois who first informed me of an interesting and elegant application of the Method of Exhaustion to calculate the area of a parabolic section determined by a parabola and a line segment AB joining two points A and B of the parabola.

Archimedes considered the “parabolic triangle ABC with vertices A, B and C where C is the point on the parabola directly below the midpoint M of the line segment AB.

Archimedes observed that area of the parabolic triangle ABC depend only on the length of the “shadow” EF of AB on the x-axis, and not on the location of EF along the x-axis. He also observed that if the two new parabolic triangles based on the segments AC and CB are constructed (i.e. the blue triangles ACJ and CBK in the figure below), then the total area of the two new triangles ACJ and CBK is one-fourth of the area of the original triangle ABC.

#### Proofs of Archimedes’ Parabolic Triangles Results

Archimedes continued this process by creating a new parabolic triangle on each of 4 line segments AJ, JC, CK and KB, with the total area of the new parabolic triangles being one-fourth of the total area of the triangles ACJ and CBK, or one-sixteenth of the area of the original parabolic triangle ABC. In this way, he “exhausted” the area of the parabolic section above the parabola and below the line segment AB. He concluded from this that the area of this parabolic section is 4/3 of the original parabolic triangle ABC. You are asked to verify this fact in Homework Problem A by using the formula for the sum of an infinite geometric series.

# Proofs of Archimedes’ Parabolic Triangles Results

## Proofs of Archimedes’ Parabolic Triangles Results

Proof: If r is the x-coordinate of the point C, there is a positive number h such that the x-coordinate of A is r – h and the x-coordinate of B is r + h C is the point of the parabola directly below the midpoint of the line segment AB. We need to prove that the area of the triangle ABC depends only on h and not on r. The coordinates of A, B and C are given by:

We will apply the following formula for the area of a triangle T with vertices at the points (a, b), (c, d), (e, f):

Apply this formula to the triangle ABC with the coordinates of A, B, and C given above and use the properties of determinants to obtain:

This shows that the area of ABC depends only on h and not on the location of the “shadow” of the interval AB on the x-axis. It also shows that the total area of the two new triangles, ACJ and CBK, is one-fourth of the area of ABC. For the “shadows” of
the triangles ACJ and CBK on the x-axis are half the length of the segment EF, so the total area of these two triangles is:

Archimedes continued this process by creating a new parabolic triangle on each of 4 line segments AJ, JC, CK and KB, with the total area of the new parabolic triangles being one-fourth of the total area of the triangles ACJ and CBK, or one-sixteenth of the area of the original parabolic triangle ABC. In this way, he “exhausted” the area of the parabolic section above the parabola and below the line segment AB. He concluded from this that the area of this parabolic section is 4/3 of the original parabolic triangle ABC. You are asked to verify this fact in Homework Problem 1 by using the formula for the sum of an infinite geometric series.

# The Method of Exhaustion and the Definite Integral in calculus

As we have already pointed out, many teachers encounter the Method of Exhaustion for the first time in their calculus course in the discussion of the definite integral.. In that course, the definite integral of a function f[x] over an interval [a, b] of the x-axis is often defined as the common limit of upper Riemann sums and lower Riemann sums associated with partitions of the interval [a, b] as the maximum length of any subinterval of the partitions approaches 0.

The discussion of the definite integral often begins with a concrete example of a function f[x] that is non-negative on the interval [a, b]. such as the function:

For a given positive integer n, we subdivide the interval [0,3] into n equal subintervals We then inscribe the region above the interval [0, 3] on the x-axis and below the graph of f[x] with n rectangles of width w[n] = 3/n and heigths equal to the value of f[x] at the left endpoint of each subinterval to obtain the lower approximation s[n]. The upper approximation S[n] is constructed in the same way but with the rectangle heights equal to the value of f[x] at the right endpoint of each successive subinterval. These upper and lower approximations are shown in the diagram below for n = 15.

The area under the graph of f[x] and above the interval [0, 3] is then defined to be the unique number that is larger than all of the areas of s[n] and smaller than the areas of S[n] for all positive integers n. You are asked to show that 9 is that number in the following:

# Just Do It!

Suppose that s[n]and S[n] are the lower and upper Riemann sums for a partition of the interval [0, 3] into n equal subintervas for the function

Show that for all positive integers n, s[n] < 9 < S[n] and that Area(S[n]) - Area(s[n]) approaches 0 as n increases without bound.

Hint: Just sum up the areas of the rectangles comprising s[n] and S[n] and simplify the results by making use of the following formulas for the sum of the squares of the first k positive integers:

For any positive integer n, the n rectangles comprising s[n] and S[n] have the same width 3/n. The successive heights of the recatngles in s[n[ are:

Therefore, the areas of a[n] and S[n] can be computed as follows:

The following formula for the sum of the squares of the first k positive integers can be verified easily with mathematical induction:

Formula (*) permits us to simply the expressions for Area(s[n]) and Area(S[n] as follows:

The fact that Area(S[n]) - Area(s[n]) approaches 0 as n increases without bound can now be seen from the formulas obtained above for Area(S[n]) and Area(s[n]).

# Assignment 2.2

Assignment Completion and Submission Directionss: Prepare a single MSWord document for Assignment 2.2. Use Equation Editor or Mathe Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create these diagrams or graphs in other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete Assignment 2.2, submit your Word file by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

In the Probelms at the end of Section 10.1.4, solve Parts a and b of Problem 1, Problem 2, 3, 5 and Parts a and b of Problem 7. Then solve Problems A listed below. Your solutions to all of these problems should be placed in a single MS Word document and submitted as Assingnment 2.2.

Problem A: As we noted in the subsection Archimedes and the Method of Exhaustion For Parabolic Sections, Archimedes determined the area of the parabolic section bounded by a parabola and a line segment AB joining two points A and B on the parabola by “exhausting” the area between the parabola and the line segment AB with a sequence of expanding inscribed polygons.

He took the first of these polygons to be the “parabolic triangle” ABC with vertices A, B and C where C is the point on the parabola directly below the midpoint M of the line segment AB. He continued by constructing triangles ACJ and CBK, based on the segments AC and CB in a similar manner to obtain the inscribed pentagon AJCKB, and so on. The figure below displays the first two steps in this exhaustion process.

Suppose that h is one-half the length of the “shadow” EF of AB on the x-axis. We proved that:

Find the area of the parabolic section above the parabola and below the segment AB by computing the limit of the sum of the areas of the inscribed polygons obtained by continuing the triangle constructions described above.

Hint: Recall the formula for the sum of a geometric series:

# Step #5

STEP #5: Complete Unit 3, Part 1: Volume formulas for polyhedra Complete and submit Part 1 of Assignment 3.1 according to the directions at the beginning of that assignment. Save that document to combine it with Part 2 of Assignment 3.1 in Step #6 below.

Comments on Step #5: Sections 10.2.1 develops the axioms for volume that are used to establish the volume formulas for geometric solids Section 10.2.2 applies these axioms to develop the volume formulas for all of the usual polyhedral solids. Cavalieri's principle, the main new axiom for volume, is discussed in the problems and through the internet-based resources of this unit.

Save a copy of your completed  Assignment 3.1 Part 1 to the Module 15 Assignments folder on your hard drive. Then submit Assignment 3.1 Part 1 by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

# Unit 3, Part 1: Volume formulas for polyhedra.

High school geometry books typically include a late chapter to the discussion and application of volume and surface area formulas for a variety of three-dimensional solids including prisms, pyramids, cones, cylinders and spheres. The surface area formulas for prisms, pyramids, cylinders and cones are justified by a "cut-and-lay-flat" procedure in which it is imagined that a paper model of the surface of the solid is cut out and laid flat as in the diagram below

.

Then the formulas for the areas of circles, rectangles, triangles and so on can be applied to produce a formula for the surface area of the given solid. Volume formulas for pyramids, prisms, cylinders and cones are usually justified in a manner similar to the Cavilieri Principle derivations that are used in our text. In such developments, the Cavilieri Principle is often not stated explicitly but rather is discussed informally in terms of slicing the solid into thin (but not infinitesimal) parallel slices.

These procedures do not work for finding formulas for surface area or volume of some solids such as a sphere. In such cases, some texts simply state the formulas as facts given without formal or informal justification. The better geometry textbooks such Geometry in the University of Chicago School Mathematics Project as provide a Cavalieri Principle in much the same way that these results are obtained in our textbook.

As mentioned before in the Teachers Perspective for Part 1 of Unit 2, most high school and beginning college students are not able to recall most of the volume and surface area formulas for familiar solids. In my opinion, this is due to the fact that most high school texts and classes emphasize routine applications of the formulas to concrete examples. The better geometry texts include problems that are a bit less superficial such as: What is the effect on the surface area and volume of a sphere if the radius is doubled? Explain your answer.", or "If the length of the sides of a box are increased by 2 inches, 3 inches and 4 inches respectively, how does that change the volume and surface area of the box? Explain your answers." Of course, a few routine formula applications are necessary to get things started, but some such 'higher level problems should be included in every assignment in my opinion.

#### Cavalieri's Principle and the Method of Exhaustion.

The Method of Exhaustion (See Part 2 of Unit 2) was used by Archimedes and others to compute the unknown area of curvilinear regions in the plane by constructing a sequence of polygonal regions whose areas are directly computable and that exhaust the given region. The area of the given region is then the limit of the values of the areas of the polygonal regions that exhaust the given region. Although we have only discussed the Method of Exhaustion for area computations, the method can be extended successfully to to the computation of volumes of regions bounded by curved surfaces.

Cavalieri's Principle is analogous to the Method of Exhaustion in the sense that it is used to compute the area and volumes of regions with curved boundaries but the two methods differ in a fundamental way. He regarded planar regions to be composed of infinitely many parallel line segments (or linear slices) and solid regions to be composed of infinitely many parallel planar regions (or planar slices). Given two such regions S and T, if one can show that each slice of S has length (or area) equal to that of the "corresponding" slice of T, then the area (or volume) of S and T are equal. Thus, Cavalieri's Principal views regions as being composed of "infinitely thin" or infinitesimal slices, while the Method of Exhaustion views such regions as "limits of approximations" by simpler regions of known area or volume. It is interesting that these two distinct approaches to the calculation of areas and volumes were reflected in the two distinct approaches that were taken by Leibniz and Newton in their development of calculus, with Leibniz following the infinitesimal approach and Newton the limit of approximations approach.

High school geometry books typically state many area and volume formulas and apply them to a variety of figures; however, few of the formulas are actually proved. While this may be reasonable for high school students, it is important that teachers have a deeper understanding of these formulas and their interdependence. Yet, college geometry courses for prospective teachers often do not provide a rigorous development of these measurement formulas Our objective in Units 2 and 3 of this module is to provide that sort of systematic development.

Part 1 of this unit will show how all of the formulas for areas of polygonal figures in the plane can be derived from the single formula for the area of a square that states that area of a square is the square of its side length. We will base these derivations on a set of axioms for the area function. Part 2 will extend and augment the area axioms to obtain formulas for the areas of circles and other regions bounded by line segments, circular arcs and other curves.

# Just Do It!

Use Cavalieri’s Principle to derive the fact that the area of any triangle is

A =1/2 (base) (altitude),

given only that the area of a rectangle is the product of its length and width.

Given a triangle ABC with base BC, the altitude is the length of the line segment AD joining A to the line through B and C. The point D either lies on the base BC or outside of that segment on the line through B and C. Consider the right triangle A’BC with the same base and equal altitude H with ABC

As can be seen from the diagram, the horizontal sections x, y and z of the right triangle A’BC (in blue), of the triangle ABC (in red) with D on the base segment BC, and the triangle ABC (in green) are all of equal length. It follows from Cavalieri’s Principle, that the areas of these three triangles are equal. But the area of the triangle A’BC is one-half the area of the rectangle with sides BC and A”B. Therefore, the area of any of the three triangles is one-half the product of the lengths of its base and its altitude.

# Assignment 3.1, Part 1

Assignment Completion and Submission Directionss: Prepare a single MSWord document for both Part 1 and Part 2 of Assignment 3.1. Use Equation Editor or Mathe Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create these diagrams or graphs in other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete both parts of Assignment 3.1, submit your Word file by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

Solve problem 5 parts b, c, and d in Section 10.2.1 (Note:  The Cavalieri's Principle mentioned in problem 5 part a is part 4 of the volume function and the reason part a is not being assigned.)

Solve problem 6 in Section 10.2.1 (Note: Part b) of Problem 6 in Section 10.2.1 should read:  b) Use Cavalieri's Principle to show that T preserves volume.) Your solutions should be places in a MS Word document that will be the Part 1 of Assignment 3.1 for this module.

Solve problems 2 and 6 in Section 10.2.2.

# Step #6

STEP #6: Complete Unit 3, Part 2: Volume and surface area formulas for cylinders, cones and other "curvy" solids. Complete and submit Part 2 of Assignment 3.1 according to the directions at the beginning of that assignment.

Comments on Step #6: Section 10.2.3.develops the volume formulas for curvilinear solids such as cylinders, cones and spheres and Section 10.3.1 develops the corresponding formulas for surface area. The internet based materials develop related history and mathematical contents including Archimedes' famous result relating both the volume and surface area of a sphere and its circumscribing cylinder in the ratio 2:3.

Complete Part 2 of Assignment 3.1. Save a copy of your completed  Assignment 3.1 Part 2 to the Module 15 Assignments folder on your hard drive. Then submit Assignment 3.1 Part 2 by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

If you are enrolled for Continuing Education Units, you have just completed the module. Congratulations!

You have been or will be sent feedback on your four assignments by e-mail and/or your graded assignments will be returned through the MTL Module Working Environment.

After all of these assignments have been completed, submitted and graded, the University of Illinois Division of Academic Outreach will send you an official letter of completion for the module. There is no transcript record for Continuing Education Units as there is with undergraduate or graduate credit, so this letter will serve as your proof of completion.

Those of you enrolled for graduate credit still need to complete an approved Final Project. Go to Step 7 to find out how to do that.

# Unit 3, Part 2

### Overview of Sections 10.2.3 and 10.3.1.

The basic tools for obtaining formulas for the volumes of non-poyhedral solids with curvilinear bases and cross-sections such as cylinders and cones and spheres are Cavalieri’s Principle, similarity of plane figures and the area formulas for the bases and cross-sections of such solids that were discussed in Unit 2. As you study the derivations of the volume formulas in Section 10.2.3, you will see these tools at work in every case.

The basic tool for obtaining the surface area formulas of non-poyhedral solids with curvilinear bases and cross-sections such as such as cylinders and cones is the same “cut-and-lay-out procedure that was used for the corresponding polyhedral solids. However, for spheres and portions of spheres cut off by planes the derivations of the formulas is substantially more difficult because they involve approximation techniques. It is truly amazing that Archimedes was able to obtain these results with purely geometric constructions and reasoning and without approximation. His ideas are not discussed in our textbook but the are outlined below

### Archimedes and the volume and surface area of a sphere.

Archimedes proved the following beautiful result about the volume and surface area of a sphere:

If a sphere is inscribed in a cylinder, then the sphere is 2/3 of the cylinder in both volume and surface area.

The following diagram explains this statement in terms of the corresponding volume and surface area formulas:

It is said that Archimedes was so pleased with this result that he requested that a cylinder and inscribed sphere be placed atop his tombstone with the ratio 2:3 engraved upon it!

Actually, Archimedes discovered and proved a more general result about the surface area of the region of the surface of a sphere sliced off by a plane. Such a region is called a spherical cap or spherical section, He showed that this surface area was equal to the area of the circle whose radius is the distance from the central point of that cap to its boundary. The following diagram shows a central cross-section of the sphere perpendicular to the plane H that slices off a spherical cap C.

Discussion of Archimedes Spherical Cap Area Formula (http://www.mathpages.com/home/kmath343.htm)

The shape of the interior of the Pantheon in Rome is that of a hemispherical dome placed atop a circular cylinder so that the imaginary sphere that contains the dome touches the center of the floor. Thus, in Archimedes’ model of a sphere inscribed in a cone, the lower half of the cylinder and the upper half of the sphere model the interior of the Pantheon. For an interesting discussion of the Pantheon and related mathematical problems see;

Mathematics and the Design of the Pantheon. (http://www.maths.adelaide.edu.au/people/pscott/place/pm20/pm20.html)

# Assignment 3.1, Part 2

Assignment Completion and Submission Directionss: Prepare a single MSWord document for both Part 1 and Part 2 of Assignment 3.1. Use Equation Editor or Mathe Type for mathematical formulas as well as inserted graphs or diagrams when approprite You can create these diagrams or graphs in other applications such as TI Link for TI graphing calculators, Geometer's Sktetchpad, or Mathematica - anything that works for you and that can be inserted in an MS Word document. The objective is to produce an assignment solutions document that is:

• technically correct and clearly described, and
• attractively displayed and organized.

Your grade on this assignment will be based on the extent to which the file that you submit meets both of these criteria.

When you complete both parts of Assignment 3.1, submit your Word file by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

## Problem Assignment

Solve Problems 2, 3, 8 and Parts a), b) and c) of Problem 9. (Note: The Hint for Problem 8 provides one approach to the solution of that problem Another simpler approach is to apply Archimedes' Formula for the surface area of a circular cap.)

Your solutions should be places in a MS Word document that will be the Part 2 of Assignment 3.1 for this module. Submit the file(s) to us at Math Teacher Link according to the instructions at the beginning of this assignment.

Congratulations! You have finished the last problem assignment and the last unit of this module. If you are enrolled in Module 15 for Continuing Education Units, you are now finished! If you are enrolled for Graduate Credit, you are now ready to propose a final project related to the module. See the Module 15 Step-by-Step Instructions for more information on the final project.

# Step #7

STEP #7: Propose and complete an approved Final Project consisting of developing a classroom unit for one of your classes based on information and techniques related to the content of this module. Final Project Topic: You are given a great deal of latitude in the choice of topic for this unit because we want the choice to reflect your teaching situation and your interests. We require only that the unit is based on the module. Of course, this module is written for teachers and your final project is for high school students, so the level and emphasis of your final project will likely be different than what is found in the module. Required Final Project Proposal: After you have selected the topic for your final project, compose an e-mail message describing in a paragraph or two how you want to develop your topic, and send it to us to us in an e-mail message at: geoform [at] mtl.math.uiuc.eduWe will respond with suggestions or ask for further explanation. Once your final project plan is approved, you can proceed with the development of the project. Details about the Final Classroom Project: Your final project should be a classroom unit that might require 1.5 - 2.5 class days to discuss in class. It must include a lesson plan, any necessary student worksheets or handouts and any electronic documents for classroom demonstrations that you develop for the project. The project should be documented well enough so that another teacher can use it without further explanation. If practical, we would also like you to teach and evaluate the unit as part of your report, but that is not an absolute requirement. The final project will count for at least one-fourth of your final grade in the module, so it should reflect a corresponding effort and time commitment.   When you have completed your Final Project, submit all documents electronically by using the Hand In button on the tab at the top of this page and using the login directions that were in your welcome letter.

You are done! The Math Teacher Link instructional staff will review and provide feedback on your assignments and Classroom Project.