Circumference and Area of Circles


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1 ircumference and Area of ircles 7 MAIN IDEA Measure and record the distance d across the circular part of an object, such as a battery or a can, through its center. Find the circumference and area of circles. New Vocabulary Place the object on a piece of paper. Mark the point where the object touches the paper on both the object and on the paper. circle center radius chord diameter circumference pi arefully roll the object so that it makes one complete rotation. Then mark the paper again. Finally, measure the distance between the marks. Math Online glencoe.com Extra Examples Personal Tutor Selfheck Quiz in What distance does represent?. Find the ratio _ for this object. d. Repeat the steps above for at least two other circular objects and compare the ratios of to d. What do you observe?. Graph the data you collected as ordered pairs, (d, ). Then describe the graph. A circle is a set of points in a plane that are the same distance from a given point in the plane, called the center. The segment from the center to any point on the circle is called the radius. A chord is any segment with both endpoints on the circle. A diameter is a chord that passes through the center. It is the longest chord. center radius (r) circumference () diameter (d) The diameter of a circle is twice its radius or d = r. The distance around the circle is called the circumference. The ratio of the circumference of a circle to its diameter is always.596. _ It is represented by the Greek letter π (pi). The numbers. and 7 are often used as approximations for π. So, _ = π. This can also be d written as = πd or.d. 5 hapter 7 Measurement: Area and Volume 5_57_07_L0_87050.indd 5 9//07 :7: PM
2 ircumference of a ircle Key oncept Words The circumference of a circle is equal to its diameter d times π, or times its radius r times π. Model d r Symbols = πd or = πr alculating with π When evaluating expressions involving π, use the π key on a calculator to obtain the most accurate result. However, using. for π will result in a close approximation. Find the ircumferences of ircles Find the circumference of each circle. Round to the nearest tenth. 9 in. = πd ircumference of a circle = π 9 Replace d with 9. = 9π This is the exact circumference. Use a calculator to find 9π. 9 [π] The circumference is about 8. inches. 7. cm = πr ircumference of a circle = π 7. Replace r with Use a calculator. The circumference is about 5. centimeters. a. b. c. 6 ft.7 m 7.5 in. A circle can be decomposed into congruent wedgelike pieces. Then the pieces can be rearranged to form a figure that resembles a parallelogram. radius Since the circle has an area that is relatively close to the area of the parallelogramshaped figure, you can use the formula for the area of a parallelogram to find the formula for the area of a circle. A = bh A = ( _ ) r A = _ ( πr )r Replace with πr. A = π r r or π r Simplify. Area of a parallelogram The base of the parallelogram is onehalf the circumference and the height is the radius. Lesson 7 ircumference and Area of ircles 5 5_57_07_L0_87050.indd 5 9//07 :7: PM
3 Area of a ircle Words The area A of a circle is equal to π times the square of the radius r. Symbols A = πr Model Key oncept r Estimation To estimate the area of a circle, square the radius and then multiply by. 8 km Find the Areas of ircles A = π r Area of a circle 5 ft A = π 8 Replace r with 8. A = π 6 Evaluate 8. This is the exact area. A 0. Use a calculator. The area is about 0. square kilometers. A = π r Area of a circle A = π (7.5) Replace r with half of 5 or 7.5. A = π 56.5 Evaluate 7.5. This is the exact area. A 76.7 Use a calculator. The area is about 76.7 square feet. d. The radius is inches. e. The diameter is 5 meters. 5 STATE PARKS Suppose you walk around the edge of the circular Point State Park fountain and estimate its circumference to be 70 feet. Based on your estimate, what is the approximate diameter of the fountain? = πd ircumference of a circle 70 = πd Replace with 70. _ 70 π = d Divide each side by π. 9.6 d Use a calculator. The diameter of the fountain is about 50 feet. f. HOME DEOR A catalog states that a circular area rug covers 9.5 square feet. What is the approximate diameter of the rug? 5 hapter 7 Measurement: Area and Volume 5_57_07_L0_87050.indd 5 9//07 :7: PM
4 Examples, (p. 5) Find the circumference of each circle. Round to the nearest tenth cm yd.5 mi Examples, (p. 5) m ft 8.5 in. Example 5 (p. 5) 7. BRAELETS When ammie finished making a friendship bracelet, it was 7.9 inches long. What was the diameter of the bracelet? For Exercises See Examples 8, 5, HELP HOMEWORK HELP Find the circumference of each circle. Round to the nearest tenth mm 8 mi 0 in. 7 km m 5. mi 7.5 ft.75 in. 6. PETS Simone purchased a circular exercise pen with a radius of.5 feet to keep her new puppy safe. Find the area inside the pen. 7. MEASUREMENT A circular table top has a radius of _ feet. A decorative trim is placed along the outside edge of the table. How long is the trim? 8. SAFETY A light in a parking lot illuminates a circular area 5 meters across. What is the area of the parking lot covered by the light? 9. BIYLES Jerrod s mountain bike has a tire diameter of 6 inches. How far will the bike travel in 00 rotations of its tires? Find the exact circumference and area of each circle. 0. The radius is.5 centimeters.. The diameter is 8.6 kilometers.. The diameter is 9 inches.. The radius is 0.6 mile. Lesson 7 ircumference and Area of ircles 55 5_57_07_L0_87050.indd 55 9//07 :7:5 PM
5 . Find the diameter of a circle if its area is square millimeters. 5. GARDENING Mr. Townes created a foot wide garden path around a circular garden. The radius of the garden is 7 feet. He wants to cover the path in stones. If Mr. Townes needs bag of stones for every 5 square feet of path, how many bags of stones will he need to cover his garden path? 7 ft ft _ Another approximate value for π is. Use this value to find the 7 circumference and area of each circle. 6. The diameter is 7 feet. 7. The radius is _ inches. 8. BAKING Joaquim is baking giant cookies for the school bake sale. They will be sold for $0 for one large cookie or $0 for three smaller cookies. Which offer is the better buy? Explain your reasoning. in. 8 in. BSHF NBMM 9. SPORTS Three tennis balls are packaged one on top of the other in a can. Which measure is greater, the can s height or circumference? Explain. 0. TREES During a construction project, barriers are placed around trees. For each inch of trunk diameter, the protection zone should have a radius of _ feet. Find the area of this zone for a tree with a trunk circumference of 6 inches. d in. d ft. GRAPHI ARTS Michael is painting a sign for a new coffee shop. On the sign, he drew a circle with a radius of feet. He then drew another circle with a radius.5 times larger. How much greater is the area of the larger circle? EXTRA PRATIE See pages 685, 706. H.O.T. Problems. FIND THE DATA Refer to the Data File on pages 6 9. hoose some data and write a realworld problem in which you would determine the area of a circle.. OPEN ENDED Draw and label a circle that has a circumference between 5 and 0 centimeters. Justify your answer.. NUMBER SENSE If the radius of a circle is halved, how will this affect its circumference and its area? What happens to the circumference and area if the radius is doubled or tripled? Explain your reasoning. (Hint: Find the circumference and area for each circle and organize the data in a table.) 56 hapter 7 Measurement: Area and Volume 5_57_07_L0_87050.indd 56 9//07 :7:7 PM
6 HALLENGE Find the area of each shaded region in. cm m m 6 cm WRITING IN 8. MATH related or different. Explain how the circumference and area of a circle are 9. In the figure below, the radius of the inscribed circle is 8 inches. What is the perimeter of square WXYZ? 0. Using the two circles shown below, circumference of circle x what is circumference of circle y? 8cm cm circle x circle y A 6π in. B 6 in. in. F π_ G _ π H _ J _ D 6π in. GEOMETRY For Exercises and, use AB with vertices A(, ), B(, ), and (, 0).. GEOMETRY Graph AB and its image after it is translated units right and unit up. (Lesson 67). GEOMETRY What are the coordinates of A B when AB is reflected over the xaxis? (Lesson 66). ART At an auction in New York ity, a.55square inch portrait of George Washington sold for $. million. About how much did the buyer pay per square inch for the portrait? (Lesson ). PREREQUISITE SKILL The price of calculators has been decreasing. A calculator sold for $5 in 995, $07 in 000, and $89 in 005. Use the look for a pattern strategy to determine the price of a similar calculator in 05 if the price decrease continues at the same rate. Lesson 7 ircumference and Area of ircles 57 5_57_07_L0_87050.indd 57 9//07 :8: PM
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