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10 LESSON S uppo tcoa c h, Ma t he ma t i c s, S t ude nte di t i on, G a de7 15 Solving Poblems with Cicles Radius, Diamete, and Cicumfeence of a Cicle Plug In Evey cicle has a cente, a adius (), a diamete (d), and a cicumfeence (C). The length of the diamete, d, of a cicle is two times the length of the adius,. The cicumfeence is the peimete of a cicle. Multiply pi ( ) by the diamete, d, to find the cicumfeence. can appoximated by If d 5 6 cm, then C cm. d C d C d If 5 3 cm, then d cm. I see! I use when I find the cicumfeence because 3.14 is the appoximate value of pi. Wods to adius a segment fom the cente of a cicle to any point on it, epesented by Know CU S S DIS diamete a segment acoss a cicle though its cente, epesented by d cicumfeence the distance aound a cicle, epesented by C pi the atio of the cicumfeence of a cicle to its diamete, C4d C 5 d o 3.14 How can you find the cicumfeence of a cicle if you know only the adius? A You can use multiplication to find the diamete of a cicle Find the diamete of the cicle. Wite the fomula fo the diamete of a cicle. Plug in the length of the adius in the fomula and multiply. Wite the length of the diamete. d5 d5 cm 3 5 The diamete measues cm cm. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC The length of the adius is half the length of the diamete! d. So 5 d 4, o _ L e ss o n 1 5 CC13_Mth_G7_SE_PDF.indd /05/13 7:06 PM

11 15 Solving Poblems with Cicles B You can use the adius to appoximate the cicumfeence of a cicle. Find the cicumfeence of the cicle. 1 Wite the fomula fo the cicumfeence using the diamete. Descibe the elationship between diamete and adius. 3 Wite a cicumfeence fomula using. 4 Plug in 3.14 fo and the length of the adius in the cicumfeence fomula. 5 Multiply. Wite the appoximate cicumfeence. PRACTICE C 5 d The length of the diamete is times the length of the adius. d 5 C 5 d o C 3 3 C 3 C m The cicumfeence is appoximately metes. Find the length of the adius o diamete when given the othe measue. I emembe! The symbol means is appoximately equal to. Rose measues the adius of he bicycle wheel to be 14 inches. She odes a new tie fo the wheel by finding its cicumfeence. Howeve, she calculates its value using 3 fo p instead of What should she expect when he tie aives? 4.5 m Cicle Fomulas can be found on p. 47. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 1 If d 5 8, then 5. If 5 15, then d 5. Find the cicumfeence of the cicle. Use 3.14 fo p. Show you wok ft d 35 in. 7 m 145

12 POWER UP Aea of a Cicle The cicumfeence of a cicle is elated to the aea of a cicle. b This octagon is divided into 8 equivalent tiangles. Its shape is close to, but not exactly, a cicle. If cut apat, the 8 tiangles can be aanged as a paallelogam. If half of the fist tiangle is cut off and moved to the end, the paallelogam becomes a ectangle. The bases of fou tiangles, o about half the cicumfeence of the cicle, make up the length of the ectangle. We can appoximate the length: l 1 (p) 5 p. The width of the ectangle is equal to the height, h, of the tiangle. The height, h, is about the same as the adius,, of the cicle, so we can appoximate the width: w. The fomula fo the aea of a ectangle is A 5 lw. If we substitute p fo l and fo w, the esult is the fomula fo the aea of a cicle. A 5 p 3, o p. I see! The aea of a cicle is p, o a little moe than thee times the adius squaed! h l w Wods to Know aea of a cicle the total numbe of squae units that fit inside the cicle; The fomula fo the aea of a cicle is A 5 p. A Juanita calculates the cicumfeence and aea fo a cicle with adius cm. She finds they ae both equal to 4p. She asks how the distance aound a cicle can be the same as the aea inside a cicle. What would you tell he? You can use the adius to appoximate the aea of a cicle. Find the aea of the cicle. 1 Wite the fomula fo the aea of a cicle. Plug in 3.14 fo p and the length of the adius in the fomula. 3 Multiply. Wite the appoximate aea. A 5 A 3 A 3 3 A in. The aea is appoximately squae inches. 5 in. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 146 Lesson 15

13 B You can use diamete to find the appoximate aea of a cicle. Find the aea of the cicle. 1 Find the adius of the cicle. 15 Solving Poblems with Cicles Always check whethe the poblem o diagam lists the diamete o the adius. Wite the fomula fo the aea of the cicle. 3 Plug in 3.14 fo p and the length of the adius in the fomula. 4 Multiply. Wite the appoximate aea. d 8 cm The adius is half the diamete, so 5 cm. A 5 A 3 A 3 3 A cm The aea is appoximately centimetes. squae Rob made an eo when he solved fo the aea of the cicle shown. What can you tell him about his wok? What is the coect answe? A 5 p 3 3 Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC PRACTICE A d 6 cm A cm Find the appoximate aea of the cicle. Use 3.14 fo. Show you wok. 1 1 mi 7 mm 3 4 d 5 m d 13 ft Cicle Fomulas can be found on p

15 15 Solving Poblems with Cicles wok togethe You can estimate the cicumfeence and aea. Then check you estimate. The adius is 9 cm. The cicumfeence is a little moe than 3 times the diamete, o twice the adius. So the cicumfeence will be about , o 54 cm. I can estimate the cicumfeence and aea to see if my answes ae easonable. 1 cm The aea is a little moe than 3 times the adius squaed. So the aea will be about , o 43 cm. C 5 p A 5 p C 5 p9 A 5 p9 C A A You can use the cicumfeence of a cicle to solve poblems. C 56.5 cm A cm Find the length of fencing needed to enclose the playgound. Use 3.14 fo p. Cicle Fomulas can be found on p Estimate the cicumfeence. Use the fomula to find the cicumfeence of the playgound. 3 Wite the length of fencing needed. 50 yd Playgound Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC The diamete is little moe than yd. The cicumfeence is a times the diamete. So the cicumfeence will be about 3, o yd. C 5 C 3 C yd The length of fencing needed to enclose the playgound is appoximately yads. Look at the fomulas fo the cicumfeence and aea of a cicle. How is the fomula fo cicumfeence elated to the fomula fo aea? 149

16 Ready to Go pactice Use the fomulas to find the cicumfeence and aea of the cicle. Use 3.14 fo p. d 0 km REMEMBER means to multiply a numbe by itself. 1 C 5 pd A 5 p C 3 A 3 10 C km A 3 3 A km 11 mi C 5 p A 5 p C 3 3 A 3 C 3 A 3 3 C mi A mi Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 150 Lesson 15

17 15 Solving Poblems with Cicles Estimate fist. Then solve. Use 3.14 fo p. 3 Ali needs a pool cove fo he cicula pool. What is the cicumfeence of the pool? 1 ft 4 Tom is planning a cicula flowe gaden fo the yad. What aea of the yad will be planted with flowes? 6 ft Solve. Use 3.14 fo p. 5 A spinkle spays wate within a 5-mete adius as it spins in a cicle. What is the appoximate aea of lawn that is wateed by the spinkle? Sketch a diagam to make sue you undestand what the poblem is descibing. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 6 A hockey puck is 7.6 centimetes in diamete. What is the cicumfeence of the puck? Make Sense of Poblems Kate keeps a cicula photogaph in a cicula pictue fame. The photogaph has a diamete of 7 cm. The fame has a diamete of 15 cm. Descibe the steps Kate can take to find the total distance aound the outside and the inside of the pictue fame. What is the total distance aound the outside and the inside of the pictue fame? Making a diagam can help you sot though the infomation given in a poblem! 151

18 Ready to Go Poblem solving READ PLAN DUCKS AMUCK A fountain, 3 ft in diamete, sits in the cente of a cicula duck pond. The pond measues 3 ft fom the fountain to its edge. What is the aea of the duck pond? What is the poblem asking you to find? the of the duck pond 3 f t 3 f t How can you solve this poblem? Find the between the aeas of the pond and the fountain. What fomula can you use? SOLVE 3 3 CHECK Find the diamete of the d ft pond and fountain. Find the adius of the pond and fountain. Find the aea of the pond and fountain. Find the adius of the fountain. Find the aea of the fountain. Subtact to find the aea of the duck pond. 5 d ft A 3 ft 5 d ft A 3 Estimate each solution to see if you answes ae easonable. The aea is a little moe than 3 times the adius squaed. The aea of the pond and fountain is about 3 3 The aea of the fountain is about 3 3 ft A 5 5 ft 5 ft. 5 ft. The aea of the pond is about 5 ft. The aea of the duck pond is appoximately ft. Duplicating any pat of this book is pohibited by law. 014 Tiumph Leaning, LLC 15 Lesson 15

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