CK-12 Geometry

by CK-12 Foundation

  Think about the area of a polygon. Imagine that you look at a square with an area of exactly Of course, the sides of the square are of length long. Now think about another polygon that is similar to the first one with a scale factor of . Every by square in the first polygon has a matching by square in the second polygon, and the area of each of these by squares is . Extending this reasoning, every of area in the first polygon has a corresponding units of area in the second polygon. So the total area of the second polygon is times the area of the first polygon.

  Warning: In solving problems it’s easy to forget that you do not always use just the scale factor. Use the scale factor in problems about lengths. But use the square of the scale factor in problems about area!

  Example 2

  Wu and Tomi are painting murals on rectangular walls. The length and width of Tomi’s wall are times the length and width of Wu’s wall.

  a. The total length of the border of Tomi’s wall is . What is the total length of the border of Wu’s wall?

  This is a question about lengths, so you use the scale factor itself. All the sides of Tomi’s wall are times the length of the corresponding side of Wu’s wall, so the perimeter of Tomi’s wall is also times the perimeter of Wu’s wall.

  The total length of the border (perimeter) of Wu’s wall is .

  b. Wu can cover his wall with quarts of paint. How many quarts of paint will Tomi need to cover her wall?

  This question is about area, since the area determines the amount of paint needed to cover the walls. The ratio of the amounts of paint is the same as the ratio of the areas (which is the square of the scale factor). Let be the amount of paint that Tomi needs.

  Tomi would need quarts of paint.

  Summary of Length and Area Relationships for Similar Polygons

  If two similar polygons are related by a scale factor of , then:

  Length: The lengths of any corresponding parts have the same ratio, . Note that this applies to sides, *Area: The ratio of the areas is . Note that this applies to areas, and any aspect of an object that

  Note: You might be able to make a pretty good guess about the volumes of similar solid figures. You’ll see more about that in Chapter 11.

  Scale Drawings and Scale Models

  One important application of similar figures is the use of scale drawings and scale models. These are two-dimensional (scale drawings) or three-dimensional (scale models) representations of real objects. The drawing or model is similar to the actual object.

  Scale drawings and models are widely used in design, construction, manufacturing, and many other fields. Sometimes a scale is shown, such as “” on a map. Other times the scale may be calculated, if necessary, from information about the object being modeled.

  Example 3

  Jake has a map for a bike tour. The scale is . He estimated that two scenic places on the tour were about apart on the map. How far apart are these places in reality?

  Each inch on the map represents a distance of . The places are about apart.

  Example 4

  Cristy’s design team built a model of a spacecraft to be built. Their model has a scale of . The actual spacecraft will be long. How long should the model be?

  Let be the length of the model.

  The model should be long.

  Example 5

  Tasha is making models of several buildings for her senior project. The models are all made with the same scale. She has started the chart below.

  a. What is the scale of the models?

  The scale is .

  b. Complete the chart below.

  Building Actual height (feet) Model height (inches)

  Sears Tower

  (Chicago)

  ?

  Empire State Building

  (New York City)

  Columbia Center

  (Seattle)

  ?

  Sears Tower: . It is high.

  Columbia Center: .

  The model should be about high.

  Why There Are No 12-Foot-Tall Giants

  Why are there no tall giants? One explanation for this is a matter of similar figures.

  Let’s suppose that there is a tall human. Compare this giant (?) to a tall person. Now let’s apply some facts about similar figures.

  The scale factor relating these two hypothetical people is . Here are some consequences of this scale factor.

  All linear dimensions of the giant would be times the corresponding dimensions of the real person. This includes height, bone length, etc.

  All area measures of the giant would be times the corresponding area measures of the real person. This includes respiration (breathing) and metabolism (converting nutrients to usable materials and energy) rates, because these processes take place along surfaces in the lungs, intestines, etc. This also includes the strength of bones, which depends on the cross-section area of the bone.

  All volume measures of the giant would be times the corresponding volume measures of the real person. (You’ll learn why in Chapter 11.) The volume of an organism generally determines its weight and mass.

  What kinds of problems do we see for our giant? Here are two severe ones.

  The giant would have bones that are times as strong, but those bones have to carry a body weight that is times as much. The bones would not be up to the task. In fact it appears that the giant’s own weight would be able to break its bones.

  The giant would have times the weight, number of cells, etc. of the real person, but only times as much ability to supply the oxygen, nutrition, and energy needed.

  Conclusion: There are no giants, and some of the reasons are nothing more, or less, than the geometry of similar figures.

  For further reading: On Being the Right Size, by J. B. S. Haldane, also available at http://irl.cs.ucla.edu/papers/right-size.html.

  Lesson Summary

  In his lesson we focused on one main point: The areas of similar polygons have a ratio that is the square of the scale factor. We also used ideas about similar figures to analyze scale drawings and scale models, which are actually similar representations of actual objects.

  Points to Consider

  You have now learned quite a bit about the lengths of sides and areas of polygons. Next we’ll build on knowledge about polygons to come to a conclusion about the “perimeter” of the “ultimate polygon,” which is the circle.

  Suppose we constructed regular polygons that are all inscribed in the same circle.

  Think about polygons that have more and more sides.

  How would the perimeter of the polygons change as the number of sides increases?

  The answers to these questions will lead us to an understanding of the formula for the circumference (perimeter) of a circle.

  Review Questions

  The figure below is made from small congruent equilateral triangles.

  congruent small triangles fit together to make a bigger, similar triangle.

  What is the scale factor of the large and small triangles?

  If the area of the large triangle is , what is the area of a small triangle? The smallest squares in the diagram below are congruent.

  What is the scale factor of the shaded square and the largest square?

  If the area of the shaded square is , what is the area of he largest square?

  Frank drew two equilateral triangles. Each side of one triangle is times as long as a side of the other triangle. The perimeter of the smaller triangle is . What is the perimeter of the larger triangle? In the diagram below, ..

  What is the scale factor of the small triangle and the large triangle?

  If the perimeter of the large triangle is , what is the perimeter of the small triangle?

  If the area of the small triangle is , write an expression for the area of the large triangle.

  If the area of the small triangle is , write an expression for the area of the trapezoid.

  The area of one square on a game board is exactly twice the area of another square. Each side
of the larger square is long. How long is each side of the smaller square?

  The distance from Charleston to Morgantown is . The distance from Fairmont to Elkins is . Charleston and Morgantown are apart on a map. How far apart are Fairmont and Elkins on the same map?

  Marlee is making models of historic locomotives (train engines). She uses the same scale for all of her models.

  The locomotive was long. The model is long.

  The Class locomotive was long.

  What is the scale of Marlee’s models?

  How long is the model of the Class locomotive?

  Review Answers

  or

  or

  or

  or equivalent

  Circumference and Arc Length

  Learning Objectives

  Understand the basic idea of a limit.

  Calculate the circumference of a circle.

  Calculate the length of an arc of a circle.

  Introduction

  In this lesson, we extend our knowledge of perimeter to the perimeter—or circumference—of a circle. We’ll use the idea of a limit to derive a well-known formula for the circumference. We’ll also use common sense to calculate the length of part of a circle, known as an arc.

  The Parts of a Circle

  A circle is the set of all points in a plane that are a given distance from another point called the center. Flat round things, like a bicycle tire, a plate, or a coin, remind us of a circle.

  The diagram reviews the names for the “parts” of a circle.

  The center

  The circle: the points that are a given distance from the center (which does not include the center or interior)

  The interior: all the points (including the center) that are inside the circle

  circumference: the distance around a circle (exactly the same as perimeter)

  radius: any segment from the center to a point on the circle (sometimes “radius” is used to mean the length of the segment and it is usually written as )

  diameter: any segment from a point on the circle, through the center, to another point on the circle (sometimes “diameter” is used to mean the length of the segment and it is usually written as )

  If you like formulas, you can already write one for a circle:

  or

  Circumference Formula

  The formula for the circumference of a circle is a classic. It has been known, in rough form, for thousands of years. Let’s look at one way to derive this formula.

  Start with a circle with a diameter of . Inscribe a regular polygon in the circle. We’ll inscribe regular polygons with more and more sides and see what happens. For each inscribed regular polygon, the perimeter will be given (how to figure that is in a review question).

  What do you notice?

  The more sides there are, the closer the polygon is to the circle itself.

  The perimeter of the inscribed polygon increases as the number of sides increases.

  The more sides there are, the closer the perimeter of the polygon is to the circumference of the circle.

  Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table below shows the results if we did this.

  Regular Polygons Inscribed in a Circle with Diameter

  Number of sides of polygon Perimeter of polygon

  As the number of sides of the inscribed regular polygon increases, the perimeter seems to approach a “limit.” This limit, which is the circumference of the circle, is approximately . This is the famous and well-known number . is an endlessly non-repeating decimal number. We often use as a value for in calculations, but this is only an approximation.

  Conclusion: The circumference of a circle with diameter is .

  For Further Reading

  Mathematicians have calculated the value of to thousands, and even millions, of decimal places. You might enjoy finding some of these megadecimal numbers. Of course, all are approximately equal to .

  The article at the following URL shows more than a million digits of the decimal for . http://wiki.answers.com/Q/What_is_the_exact_value_for_Pi_at_this_moment

  Tech Note - Geometry Software

  You can use geometry software to continue making more regular polygons inscribed in a circle with diameter and finding their perimeters.

  Can we extend this idea to other circles? First, recall that all circles are similar to each other. (This is also true for all equilateral triangles, all squares, all regular pentagons, etc.)

  Suppose a circle has a diameter of .

  The scale factor of this circle and the one in the diagram and table above, with diameter , is , , or just .

  You know how a scale factor affects linear measures, which include perimeter and circumference. If the scale factor is , then the perimeter is times as much.

  This means that if the circumference of a circle with diameter is , then the circumference of a circle with diameter is .

  Circumference Formula

  Let be the diameter of a circle, and the circumference.

  Example 1

  A circle is inscribed in a square. Each side of the square is long. What is the circumference of the circle?

  Use . The length of a side of the square is also the diameter of the circle.

  Note that sometimes an approximation is given using . In this example the circumference is using that approximation. An exact is given in terms of (leaving the symbol for in the answer rather than multiplying it out. In this example the exact circumference is .

  Arc Length

  Arcs are measured in two different ways.

  Degree measure: The degree measure of an arc is the fractional part of a complete circle that the arc is.

  Linear measure: This is the length, in units such as centimeters and feet, if you traveled from one end of the arc to the other end.

  Example 2

  Find the length of .

  = . The radius of the circle is .

  Remember, is the measure of the central angle associated with .

  is of a circle. The circumference of the circle is

  . The arc length of is .

  In this lesson we study the second type of arc measure—the measure of an arc’s length. Arc length is directly related to the degree measure of an arc.

  Suppose a circle has:

  circumference

  diameter

  radius

  Also, suppose an arc of the circle has degree measure .

  Note that is the fractional part of the circle that the arc represents.

  Arc length

  Lesson Summary

  This lesson can be summarized with a list of the formulas developed.

  Radius and diameter:

  Circumference of a circle:

  Points to Consider

  After perimeter and circumference, the next logical measure to study is area. In this lesson, we learned about the perimeter of a circle (circumference) and the arc length of a sector. In the next lesson we’ll learn about the areas of circles and sectors.

  Review Questions

  Prove: The circumference of a circle with radius is

  The Olympics symbol is five congruent circles arranged as shown below. Assume the top three circles are tangent to each other.

  Brad is tracing the entire symbol for a poster. How far will his pen point travel?

  A truck has tires that measure from the center of the wheel to the outer edge of the tire. How far forward does the truck travel every time a tire turns exactly once?

  How many times will the tire turn when the truck travels ? .

  The following wire sculpture was made from two perpendicular segments that intersect each other at the center of a circle.

  If the radius of the circle is , how much wire was used to outline the shaded sections?

  The circumference of a circle is . What is the radius of the circle?

  A gear with a radius of inches turns at a rate of R
PM (revolutions per minute). How far does a point on the edge of the pulley travel in one second?

  A center pivot irrigation system has a boom that is long. The boom is anchored at the center pivot. It revolves around the center pivot point once every three days. How far does the tip of the boom travel in one day?

  The radius of Earth at the Equator is about . Belem (in Brazil) and the Galapagos Islands (in the Pacific Ocean) are on (or very near) the Equator. The approximate longitudes are Belem, , and Galapagos Islands, . What is the degree measure of the major arc on the Equator from Belem to the Galapagos Islands?

  What is the distance from Belem to the Galapagos Islands on the Equator the “long way around?”

  A regular polygon inscribed in a circle with diameter has sides. Write a formula that expresses the perimeter, , of the polygon in terms of . (Hint: Use trigonometry.)

  The pulley shown below revolves at a rate of RPM.

  How far does point travel in one hour?