1. Copyright © Cengage Learning. All rights reserved. Geometry as Measurement CHAPTER 10

2. Copyright © Cengage Learning. All rights reserved. SECTION 10.2 Perimeter and Area

3. 3 What Do You Think? Can two shapes with different perimeters have the same area? Why or why not? What does pi mean?

4. 4 Perimeter

5. 5 Let us begin with the concept of perimeter, essentially the distance around an object. Many practical applications of perimeter involve surrounding an object—for example, fencing a yard or running a baseboard around the base of a room. We will look first at distances around circles.

6. 6 Circumference and 

7. 7 When we determine the distances around figures and objects, sometimes the path is not a straight line but rather is a circle. You may remember a formula involving the distance around a circle, or the circumference, and that it involves . The value of  is 3.14 (to two decimal places).

8. 8 Circumference and  In one sense,  is a ratio—that is,  is the ratio of the circumference to the diameter of any circle. If we could precisely measure the circumference and diameter of any circle and then divide the circumference of the circle by its diameter, we would always get . If we call the circumference C and the diameter d, we have  =. Thus we have the formulas C =  d and C = 2  r

9. 9 Circumference and  In Figure 10.5, C is the center of the circle, AC, BC, and DC are radii (radii is the plural of radius), and AD is a diameter. Figure 10.5

10. 10 Circumference and  There is another way to think about . Imagine placing a string around the circumference of the circle in Figure 10.5 and then straightening out that string. If the diameter of the circle is d, then the length of the string is about 3.14d. That is, if we unwrap the circumference, its length will always be about 3.14 times the length of the diameter.

11. 11 Investigation A – What is the Length of the Arc? In Figure 10.6, the radius of the circle is 3 centimeters and the measure of angle BOC is 42 degrees. How long is arc BC? Figure 10.6

12. 12 Investigation A – Discussion Strategy 1: Make A Simpler Problem What if the angle had been 90 degrees? Can you solve that problem? If the angle had been 90 degrees, then the arc would simply have been one-fourth of the circumference of the circle. Because 42 does not divide 360 evenly, we don’t have an easy fraction. However, we can multiply the circumference of the circle by.

13. 13 Investigation A – Discussion Strategy 2: Use A Proportion We could also solve this problem using a proportion. Can you make the proportion? Arc BC is a fraction of the circumference of the circle. Similarly, 42 degrees is a fraction of 360 degrees, the number of degrees in a circle. From the ratio construct of fractions, these two ratios are equal. cont’d

14. 14 Investigation A – Discussion The circumference of the circle is  d, which equals approximately 18.84 inches, so we can set up the following proportion, because the two part: whole ratios are equal: When we solve the equation, we find that the length of arc BC is about 2.2 inches. cont’d

15. 15 Area

16. 16 Area Questions about area generally deal with “how much” it takes to cover an object—for example, how much fertilizer to cover a lawn, how much material to cover a bed. In order to answer area questions, we have to select an appropriate unit, and thus the answer takes the form of how many of those units. However, the units for perimeter and area are not the same.

17. 17 Area For example, if we have a 20-foot by 10-foot garden, we say that we need 60 feet of fence to surround the garden, but we would say that the area of the garden is 200 square feet. The need for units and the difference between units for perimeter, area, and volume are generally not well understood by students and therefore are worth emphasizing more than once.

18. 18 Area Base and height: Most students use the more common terms length and width, so let’s take a moment to examine what base and height mean, because they will become important when we investigate other figures for which the terms length and width are not appropriate. Any polygon can be rotated so that its bottom side will be parallel to the bottom of the paper. Thus any side can be taken as the base of the polygon.

19. 19 Area For example, we can rotate the triangle in Figure 10.7 so that any one of its sides is the base. Which one we choose is generally an arbitrary decision. The height of a polygon is the distance from the side chosen as the base to the point farthest away, measured along a line perpendicular to the base. See Figure 10.7 for an illustration of this notion with triangles. Figure 10.7

20. 20 Area Thinking about area by looking at rectangles: Let us focus on some more subtle aspects of the concept of area. Consider the diagrams in Figure 10.8 of two pieces of plastic sheeting that I use to cover two different woodpiles in my back yard. One piece is 10 feet by 3 feet, and the other is 6 feet by 5 feet. Figure 10.8

21. 21 Area When I ask young children if the sheets are the same size or if one is bigger, they generally say that the 10-foot by 3-foot sheet is bigger. Older children and adults know that the areas of the sheets are equal, 30 square feet. We can demonstrate this by covering each figure with 30 squares, each of which is 1 foot by 1 foot (Figure 10.9). Figure 10.9

22. 22 Area Understanding the area formula for parallelograms: The formulas for many common figures are striking, both in their simplicity and in their connection to one another. The formula for determining the area of a parallelogram is connected to the formula for determining the area of a rectangle. The diagram at the right in Figure 10.11 illustrates the connection. Figure 10.11

23. 23 Area If we cut the triangle from the parallelogram and reconnect it at the right-hand side, we have transformed the parallelogram into a rectangle. We have not added or subtracted any area, so the areas of the parallelogram and the rectangle must be equal. From Figure 10.11, you can see that the base of the parallelogram is congruent to the base of the rectangle and that the height of the parallelogram is congruent to the height of the rectangle.

24. 24 Area Because we know that the areas of the two are equal, we can now state that the area of any parallelogram is equal to the product of its base and its height; that is, A = bh (Figure 10.12). Figure 10.12

25. 25 Area Understanding the area formula for triangles: Consider triangle MAN (Figure 10.13). If we make a congruent copy of this triangle and move that triangle into place as shown (that is, by rotating it 180 degrees), we form a parallelogram. Figure 10.13

26. 26 Area The base of the parallelogram and the base of the triangle are identical, and so are the heights. If the area of the parallelogram = base  height, then (since the area of the two triangles is equal to the area of the parallelogram), (base  height), or A = b  h.

27. 27 Area Understanding the area formula for trapezoids: Can you now apply what you have learned about the areas of these figures to determine the formula for the area of a trapezoid (Figure 10.14)? There are several ways to derive the formula for the area of a trapezoid. We will discuss two that connect to our preceding discussions. Figure 10.14

28. 28 Area First, we can draw a diagonal of the trapezoid, which cuts (decomposes) it into two triangles. We know how to find the areas of these triangles: The areas are ah and bh. The sum of the areas of the two triangles is equal to the area of the trapezoid, so the area A of the trapezoid is ah + bh, or A = (a + b)h. (Figure 10.15). Figure 10.15

29. 29 Area Figure 10.16 shows another method for finding the area of a trapezoid. Figure 10.16

30. 30 Area If we construct a congruent trapezoid and connect it (via a translation and a rotation) to the original trapezoid, we have a parallelogram, and the area of the parallelogram is equal to the product of its base and its height; that is, the area is equal to (a + b)h. Because the area of the trapezoid is equal to one-half the area of this parallelogram, the area of the trapezoid = (a + b)h.

31. 31 Investigation B – Converting Units of Area Converting area units is more complicated than converting linear units, and thus an investigation will help you to own this idea. Malik and Lee have decided to pull up their old, shabby carpet and buy new carpet. The room measures 12 feet by 15 feet, so the area is 180 square feet. However, when they go to the carpet store, they find that the prices are in square yards. How many square yards is their floor?

32. 32 Investigation B – Discussion A common answer is 60 square yards—that is, 180 square feet divided by 3, because there are 3 yards in 1 foot. However, that is wrong. The diagram at the right illustrates why you must divide by 9. Even though there are 3 feet in 1 yard, there are 9 square feet in 1 square yard.

33. 33 Pythagorean Theorem

34. 34 Pythagorean Theorem Although the relationship between the lengths of the sides of a right triangle had been known long before Pythagoras, it was Pythagoras who proved that for any right triangle, the sum of the squares of the lengths of the two sides (say, a and b) is equal to the square of the length of the hypotenuse (say, c); that is, a 2 + b 2 = c 2 [Figure 10.17(a)]. Figure 10.17(a)

35. 35 Pythagorean Theorem It is important to note that for the Greeks, this was not an algebraic relationship but rather a geometric relationship. That is, if you make a square whose sides are congruent to the length of side a, a square whose sides are congruent to the length of side b, and a square whose sides are congruent to the length of the hypotenuse c, then the sum of the areas of the two smaller squares will be equal to the area of the larger square [Figure 10.17(b)]. Figure 10.17(b)

36. 36 Investigation C – Using Pythagorean Theorem We can use the Pythagorean theorem to determine the length of a lake. In Figure 10.18, a right triangle has been created in such a way that the length of the lake is part of the length of one side of the triangle. Figure 10.18

37. 37 Investigation C – Using Pythagorean Theorem The lengths of AB, BC, and CD have been measured, because they are on land. The triangle has been carefully made so that all lines are straight lines, so that point D lies on side AC, and so that angle C is 90 degrees. What is the length of the lake? cont’d

38. 38 Investigation C – Discussion Let us discuss two different strategies and then discuss problem solving. Strategy 1: Let x be the length of the pond. Using the Pythagorean theorem, we have (x + 45) 2 + 30 2 = 92 2 We now have a quadratic equation, which we can solve using the quadratic formula: x 2 + 90x + 2025 + 900 = 8464

39. 39 Investigation C – Discussion x 2 + 90x – 5539 = 0 x  42 feet Strategy 2: Looking at the problem from a different perspective, we can find the length of the base of the triangle (AC) rather easily, and then we can subtract 45 feet from the base to find the length of the lake. AC 2 + BC 2 = AB 2 AC 2 + 900 = 8464 cont’d

40. 40 Investigation C – Discussion AC 2 = 7564 AC  87 feet (to the nearest whole foot) Thus AD  42 feet (87 – 45). This investigation illustrates the idea of examining a problem from different perspectives. The second solution is quicker and simpler, although the first solution is not difficult for someone who is adept at algebra. cont’d

41. 41 Investigation D – Understanding the Area Formula for Circles The last area formula that we will consider now is that for the area of a circle. The purpose of this investigation is in the realm of “number sense,” helping you to see why A  3.14r 2 (and therefore A =  r 2 ) makes sense.

42. 42 Investigation D – Understanding the Area Formula for Circles Consider the circle in Figure 10.19(a), whose radius is r inches. In Figure 10.19(b), I have circumscribed a square around the circle. What is the area of the square? Figure 10.19(a) Figure 10.19(b) cont’d

43. 43 Investigation D – Discussion If the radius of the circle is r inches, then the length of each side of the square is 2r, and thus the area of the square is (2r)(2r) = 4r 2. The area of the circle thus is clearly less than 4 times r 2. We can use our spatial sense to estimate that the circle covers about as much space as the square and thus approximate the area of the circle as 3r 2, or we can place a grid over the figure and determine what fraction of the square is covered by the circle.

44. 44 Investigation D – Discussion In this case, we get a more accurate estimate of the area of the circle [see Figure 10.19(c)]. Figure 10.19(c) cont’d

45. 45 Investigation E – A 16-Inch Pizza Versus an 8-Inch Pizza Let’s say you are going out to have pizza with several friends. You are thinking of getting one large pizza—let’s say its diameter is 16 inches. However, some people are vegetarians, and so you decide to get two little pizzas, each of which is 8 inches in diameter. Suddenly someone asks, “Are we getting the same amount of pizza if we get two little pizzas instead of one large one?”

46. 46 Investigation E – Discussion This is generally one of the most amazing investigations in this book because so many people are so surprised. To determine the area, we can use the area formulas: If the large pizza has a diameter of 16 inches, it has a radius of 8 inches. Therefore, its area is  (8) 2 = 64   201 square inches.

47. 47 Investigation E – Discussion If the small pizza has a diameter of 8 inches, it has a radius of 4 inches. Therefore, its area is  (4) 2 = 16 . Therefore, the area of two small pizzas is 32 , or approximately 100.5 square inches. Not only are two 8-inch pizzas not equal to one 16-inch pizza, but it takes four 8-inch pizzas to have the same area as one 16-inch pizza. cont’d

48. 48 Investigation F – How Big Is the Footprint? The area of most objects cannot be determined by a formula. For example, biologists and environmentalists want to know the total surface area of the leaves of a tree in order to determine the amount of oxygen the tree produces. What if we wanted to measure the area of another irregularly shaped object, such as the footprint in Figure 10.20? Figure 10.20

49. 49 Investigation F – Discussion There are many possible strategies. Several are briefly described below. Strategy 1: Use Graph Paper This is one of the simpler strategies. There are two issues worth pursuing: (1) what to do with the partially filled squares, and (2) the advantages and disadvantages of graph paper with smaller squares.

50. 50 Investigation F – Discussion Strategy 2: Draw Rectangles and Triangles We can partition the footprint into rectangles and triangles. One advantage of this over the previous strategy is that in this case, one rectangle will account for a large portion of the print. One disadvantage is that there is room for error, depending on how the rectangles and triangles are made. This procedure could also become a bit tedious. cont’d

51. 51 Investigation F – Discussion Strategy 3: Draw Trapezoids We can partition the footprint into trapezoids. This strategy arises from the realization that trapezoids are easily broken into rectangles and triangles; also, this method involves less computation than the previous method. I can approximate the area of the footprint with four trapezoids. cont’d

52. 52 Investigation F – Discussion Strategy 4: Find the Perimeter We can trace the figure with string to get the perimeter of the print, then make a rectangle with the string and compute the area of the rectangle. There is a flaw in the reasoning behind this strategy. If you do not see this flaw, revisit this problem after doing the next investigation. cont’d

53. 53 Investigation F – Discussion Strategy 5: Weigh It We can trace this print onto a piece of thick posterboard and cut it out. We can then cut out another piece of posterboard in the shape of a rectangle. cont’d

54. 54 Investigation F – Discussion The ratio of the area of the footprint to the area of the rectangle will be equal to the ratio of the weight of the footprint to the weight of the rectangle. Therefore, we can use the following proportion: cont’d

55. 55 Investigation G – Making a Fence with Maximum Area Joshua has decided to build a fence around his garden, and he buys 200 feet of chicken wire. After he gets home and unrolls the wire, he finds to his surprise that there are many different size gardens that he can make with 200 feet of fencing. He decides that he wants to get the largest garden from 200 feet of fencing. If he wants a garden that is rectangular in shape, what are the dimensions of that garden?

56. 56 Investigation G – Discussion Strategy 1: Use Guess-Check-Revise Table 10.9

57. 57 Investigation G – Discussion Strategy 2: Reason and Make a Model A very tactile way to “feel” this investigation is to cut a piece of string and tie the ends together. Now, using your two thumbs and two forefingers, make as skinny a rectangle as you can (Figure 10.21). Figure 10.21 cont’d

58. 58 Investigation G – Discussion Now slowly move your thumbs and forefingers apart—that is, decrease the length and increase the width. What seems to be happening to the area? Why must it stop increasing when the length and width are equal? cont’d

59. 59 Investigation G – Discussion Strategy 3: Be Creative and Adventurous What if we made the garden in the shape of a circle? Because the circumference of the circle is 200 feet, the radius of the circle will be cont’d

60. 60 Investigation G – Discussion Now that we know the radius, we can find the area: A =  r 2  3.14(31.8) 2 = 3175 square feet Compare the area of the circular garden to that of the square garden: The circular shape will give Joshua more than 25 percent more area. cont’d