Area and Circumference


 Jonas Holland
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1 4.4 Area and Circumference 4.4 OBJECTIVES 1. Use p to find the circumference of a circle 2. Use p to find the area of a circle 3. Find the area of a parallelogram 4. Find the area of a triangle 5. Convert square units In Section 4.2, we again looked at the perimeter of a straightedged figure. The distance around the outside of a circle is closely related to this concept of perimeter. We call the perimeter of a circle the circumference. Definitions: Circumference of a Circle The circumference of a circle is the distance around that circle. Radius O d Diameter Circumference Figure 1 Let s begin by defining some terms. In the circle of Figure 1, d represents the diameter. This is the distance across the circle through its center (labeled with the letter O, for origin). The radius r is the distance from the center to a point on the circle. The diameter is always twice the radius. It was discovered long ago that the ratio of the circumference of a circle to its diameter always stays the same. The ratio has a special name. It is named by the Greek letter p (pi). 22 Pi is approximately, or 3.14 rounded to two decimal places. We can write the following 7 formula. NOTE The formula comes from the ratio C d p 4.5 ft Figure 2 Rules and Properties: C pd (1) Example 1 Finding the Circumference of a Circle Formula for the Circumference of a Circle A circle has a diameter of 4.5 ft, as shown in Figure 2. Find its circumference, using 3.14 for p. If your calculator has a p key, use that key instead of a decimal approximation for p. 341
2 342 CHAPTER 4 DECIMALS NOTE Because 3.14 is an approximation for pi, we can only say that the circumference is approximately 14.1 ft. The symbol means approximately. By Formula (1), C pd ft 14.1 ft (rounded to one decimal place) CHECK YOURSELF 1 NOTE If you want to approximate p, you needn t worry about running out of decimal places. The value for pi has been calculated to over 100,000,000 decimal places on a computer (the printout was some 20,000 pages long). A circle has a diameter of 3 1 inches (in.). Find its circumference. 2 Note: In finding the circumference of a circle, you can use whichever approximation for pi you choose. If you are using a calculator and want more accuracy, use the p key. There is another useful formula for the circumference of a circle. NOTE Because d 2r (the diameter is twice the radius) and C pd, we have C p(2r), or C 2pr. Rules and Properties: Formula for the Circumference of a Circle C 2pr (2) Example 2 Finding the Circumference of a Circle A circle has a radius of 8 in., as shown in Figure 3. Find its circumference using 3.14 for p. 8 in. Figure 3 From Formula (2), C 2pr in in. (rounded to one decimal place) CHECK YOURSELF 2 Find the circumference of a circle with a radius of 2.5 in.
3 AREA AND CIRCUMFERENCE SECTION Sometimes we will want to combine the ideas of perimeter and circumference to solve a problem. Example 3 Finding Perimeter We wish to build a wroughtiron frame gate according to the diagram in Figure 4. How many feet (ft) of material will be needed? NOTE The distance around the semicircle is 1 2 pd. 5 ft 4 ft Figure 4 The problem can be broken into two parts. The upper part of the frame is a semicircle (half a circle). The remaining part of the frame is just three sides of a rectangle. NOTE Using a calculator with a p key, 1 2 p 5 Circumference (upper part) ft 7.9 ft 2 Perimeter (lower part) ft Adding, we have ft We will need approximately 20.9 ft of material. CHECK YOURSELF 3 Find the perimeter of the following figure. 6 yd 8 yd The number pi (p), which we used to find circumference, is also used in finding the area of a circle. If r is the radius of a circle, we have the following formula.
4 344 CHAPTER 4 DECIMALS Rules and Properties: Formula for the Area of a Circle NOTE This is read, Area equals pi r squared. You can multiply the radius by itself and then by pi. A pr 2 (3) Example 4 Find the Area of a Circle A circle has a radius of 7 inches (in.) (see Figure 5). What is its area? 7 in. Figure 5 Use Formula (3), using 3.14 for p and r 7 in. A 3.14 (7 in.) in. 2 Again the area is an approximation because we use 3.14, an approximation for p. CHECK YOURSELF 4 Find the area of a circle whose diameter is 4.8 centimeters (cm). Remember that the formula refers to the radius. Use 3.14 for p, and round your result to the nearest tenth of a square centimeter. Two other figures that are frequently encountered are parallalograms and triangles. B C h A b D Figure 6 In Figure 6 ABCD is called a parallelogram. Its opposite sides are parallel and equal. Let s draw a line from D that forms a right angle with side BC. This cuts off one corner of the parallelogram. Now imagine that we move that corner over to the left side of the figure, as shown. This gives us a rectangle instead of a parallelogram. Because we haven t changed the area of the figure by moving the corner, the parallelogram has the same area as the rectangle, the product of the base and the height.
5 AREA AND CIRCUMFERENCE SECTION Rules and Properties: Formula for the Area of a Parallelogram A b h (4) Example 5 Finding the Area of a Parallelogram A parallelogram has the dimensions shown in Figure 7. What is its area? 1.8 in. 3.2 in. Figure 7 Use Formula (4), with b 3.2 in. and h 1.8 in. A b h 3.2 in. 1.8 in in. 2 CHECK YOURSELF 5 If the base of a parallelogram is 3 1 in. and its height is 1 1 in., what is its area? 2 2 Another common geometric figure is the triangle. It has three sides. An example is triangle ABC in Figure 8. B D A h b Figure 8 C b is the base of the triangle. h is the height, or the altitude, of the triangle. Once we have a formula for the area of a parallelogram, it is not hard to find the area of a triangle. If we draw the dotted lines from B to D and from C to D parallel to the sides of the triangle, we form a parallelogram. The area of the triangle is then onehalf the area of the parallelogram [which is b h by Formula (4)].
6 346 CHAPTER 4 DECIMALS Rules and Properties: Formula for the Area of a Triangle A 1 2 b h (5) Example 6 Finding the Area of a Triangle A triangle has an altitude of 2.3 in., and its base is 3.4 in. (see Figure 9). What is its area? 2.3 in. 3.4 in. Figure 9 Use Formula (5), with b 3.4 in. and h 2.3 in. A 1 2 b h in. 2.3 in in.2 2 CHECK YOURSELF 6 A triangle has a base of 10 feet (ft) and an altitude of 6 ft. Find its area. Sometimes we will want to convert from one square unit to another. For instance, look at 1 yd 2 in Figure yd = 3 ft 1 yd = 9 ft 1 yd = 3 ft NOTE Originally the acre was the area that could be plowed by a team of oxen in a day! Figure 10 The table below gives some useful relationships. Square Units and Equivalents 1 square foot (ft 2 ) 144 square inches (in. 2 ) 1 square yard (yd 2 ) 9 ft 2 1 acre 4840 yd 2 43,560 ft 2
7 AREA AND CIRCUMFERENCE SECTION Example 7 Converting Between Feet and Yards in Finding Area A room has the dimensions 12 ft by 15 ft. How many square yards of linoleum will be needed to cover the floor? 12 NOTE We first find the area in square feet, then convert to square yards. A 12 ft 15 ft 180 ft yd ft 2 9 ft 2 20 yd 2 CHECK YOURSELF 7 A hallway is 27 ft long and 4 ft wide. How many square yards of carpeting will be needed to carpet the hallway? Example 8 illustrates the use of a common unit of area, the acre. Example 8 Converting Between Yards and Acres in Finding Area A rectangular field is 220 yd long and 110 yd wide. Find its area in acres. A 220 yd 110 yd 24,200 yd ,200 yd acres CHECK YOURSELF 8 acre yd 2 A proposed site for an elementary school is 220 yd long and 198 yd wide. Find its area in acres.
8 348 CHAPTER 4 DECIMALS CHECK YOURSELF ANSWERS 1. C 11 in. 2. C 15.7 in. 3. P 31.4 yd cm 2 5. A A 1 10 ft 6 ft yd acres 2 in in in. 3 2 in. = 30 ft in.2
9 Name 4.4 Exercises Section Date Find the circumference of each figure. Use 3.14 for p, and round your answer to one decimal place ft 5 ft ANSWERS in ft In exercises 5 and 6, use 22 7 for p, and find the circumference of each figure in ft Find the perimeter of each figure. (The curves are semicircles.) Round answers to one decimal place ft 7 ft ft 7 ft 1 in. 3 in. 10 in. 349
10 ANSWERS Find the area of each figure. Use 3.14 for p, and round your answer to one decimal place in. 12 ft yd 8 ft 20. In exercises 15 and 16, use for p, and find the area of each figure yd in. Find the area of each figure ft 7 ft 8 in yd 3 yd 5 in. 4 in. 7 in. 350
11 ANSWERS ft 8 ft 6 ft 11 ft in. 13 yd in. 13 yd ft 4 ft 6 yd 5 ft 7 yd 2 yd Solve the following applications. 27. Jogging. A path runs around a circular lake with a diameter of 1000 yards (yd). Robert jogs around the lake three times for his morning run. How far has he run? 28. Binding. A circular rug is 6 feet (ft) in diameter. Binding for the edge costs $1.50 per yard. What will it cost to bind around the rug? 29. Lawn care. A circular piece of lawn has a radius of 28 ft. You have a bag of fertilizer that will cover 2500 ft 2 of lawn. Do you have enough? 351
12 ANSWERS Cost. A circular coffee table has a diameter of 5 ft. What will it cost to have the top refinished if the company charges $3 per square foot for the refinishing? Cost. A circular terrace has a radius of 6 ft. If it costs $1.50 per square foot to pave the terrace with brick, what will the total cost be? 32. Area. A house addition is in the shape of a semicircle (a half circle) with a radius of 9 ft. What is its area? 33. Amount of material. A TetraKite uses 12 triangular pieces of plastic for its surface. Each triangle has a base of 12 inches (in.) and a height of 12 in. How much material is needed for the kite? 34. Acreage. You buy a square lot that is 110 yd on each side. What is its size in acres? 35. Area. You are making rectangular posters 12 by 15 in. How many square feet of material will you need for four posters? 36. Cost. Andy is carpeting a recreation room 18 feet (ft) long and 12 ft wide. If the carpeting costs $15/yd 2, what will be the total cost of the carpet? 37. Acreage. A shopping center is rectangular, with dimensions of 550 by 440 yd. What is its size in acres? 38. Cost. An Aframe cabin has a triangular front with a base of 30 ft and a height of 20 ft. If the front is to be glass that costs $3 per square foot, what will the glass cost? 352
13 ANSWERS Find the area of the shaded part in each figure. Round your answers to one decimal place Semicircle 3 ft ft 5 ft ft ft 10 in ft 43. Papa Doc s delivers pizza. The 8inch (in.)diameter pizza is $8.99, and the price of a 16in.diameter pizza is $ Write a plan to determine which is the better buy. 10 in The distance from Philadelphia to Sea Isle City is 100 miles (mi). A car was driven this distance using tires with a radius of 14 in. How many revolutions of each tire occurred on the trip? 45. Find the area and the circumference (or perimeter) of each of the following: (a) a penny (b) a nickel (c) a dime (d) a quarter (e) a halfdollar (f) a silver dollar (g) a Susan B. Anthony dollar (h) a dollar bill (i) one face of the pyramid on the back of a $1 bill. 46. An indoor track layout is shown below. 20 m 7 m How much would it cost to lay down hardwood floor if the hardwood floor costs $10.50 per square meter? 353
14 ANSWERS What is the effect on the area of a triangle if the base is doubled and the altitude is cut in half? Create some examples to demonstrate your ideas How would you determine the crosssectional area of a Douglas fir tree (at, say, 3 ft above the ground), without cutting it down? Use your method to solve the following problem: 50. If the circumference of a Douglas fir is 6 ft 3 in., measured at a height of 3 ft above the ground, compute the crosssectional area of the tree at that height. 49. What happens to the circumference of a circle if you double the radius? If you double the diameter? If you triple the radius? Create some examples to demonstrate your answers. 50. What happens to the area of a circle if you double the radius? If you double the diameter? If you triple the radius? Create some examples to demonstrate your answers. Answers ft in in ft ft in yd ft yd 2 8 yd ft in ft yd 29. Yes; area ft $ in ft acres ft ft Doubled; doubled; tripled 354
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