16.1 Justifying Circumference and Area of a Circle

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Name Class Date 16.1 Justifying Circumference and Area of a Circle

Essential Question: How can you justify and use the formulas for the circumference and area Resource of a circle? Locker

Explore Justifying the Circumference Formula

To find the circumference of a given circle, consider a regular polygon that is inscribed in the circle. As you increase the number of sides of the polygon, the perimeter of the polygon gets closer to the circumference of the circle.

Inscribed Inscribed Inscribed pentagon hexagon octagon

Let circle O be a circle with center O and radius r. Inscribe a regular n–gon in circle O and draw radii from O to the vertices of the n-gon.

r 1 2 r

A x M B

_ _ _ Let AB be one side of the n-gon. Draw OM , the segment from O to the midpoint of AB .

A Then △AOM ≅ △BOM by .

B So, ∠1 ≅ ∠2 by . © Houghton Mifflin Houghton © Company Harcourt Publishing C There are n triangles, all congruent to △AOB, that surround point O and fill the n-gon.

Therefore, m∠AOB = and m∠1 = .

Module 16 851 Lesson 1 © Houghton Mifflin Harcourt Publishing Company Lesson Lesson 1 . )

are n 180 ____ ( BOM sin sin △ n and . AOM . . △ form a linear pair, these pair, a linear form –gon. What happens to to happens What –gon. ? n r . So ° π

1 from above gives OMB 2 ∠ ∠ . is r = x

C gon ever equal the circumference of of equal ever the circumference gon 852 and .

n- r x _ gets larger? gets

OMA

–gon should include the factor the factor include should –gon . as is ∠ 1

n )

r ° Y

x 180 ____ as ( -gon in terms of of in terms -gon -gon in terms of of in terms -gon )

n x gets larger? gets gets larger? gets

180 ___ sin sin n n found in Step D. in Step found ( x x sin sin

x by CPCTC, and and CPCTC, by -gon in terms of of in terms -gon , because n x length of hypotenuse length of length of opposite leg opposite length of __

is 2 OMB =

∠ 1

. _ AB ∠ 1 and substituting the expression for m for the expression substituting 1 and

≅

∠ , sin sin r OMA

is very large, does the perimeter of the does the very perimeter is of large, n ∠ = sin AOM

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the the perimeter of for wrote you the expression Look at the the perimeter of for expression Your the the perimeter of express Now Since Since Reflect the perimeter of The the the perimeter of means This Substitute the expression for So, Module Module 16 2. 1.

   length of The In 

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© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Nikada/ iStockPhoto.com The circumference is about 125.6feet. C C oue 16 Module 4. Your Turn 3. Reflect C The diameter radius the is 2feet, so ininches is B 20 Diameter A Explain1

C Use formula the Example 1 AFerris has wheel adiameter of 40feet. What is its A pottery wheel has wheel adiameter Apottery of 2feet. What is its circumference? Use 3.14for compare with circumference the of circle? smaller the Explain. Use 3.14for diameter? Round to nearest the tenth of afoot. The circumference of atree is 20feet. What is its Discussion circumference? Use 3.14for C C C

≈ = ≈ = = = = 125.6 2

2 2 2 Find circumference the indicated. ( π π π 40 3.14 ( r r 20 = = =

∙ Suppose you double radius the of acircle. How circumference the does of larger this circle )

) π 2 2 r

( . 20 r r ) in.

Applying theCircumference Formula C

∙ =

2 π r to find the circumference. tothe find π . 853 r

= 5.

tenth of afoot. Use 3.14for feet. What is its diameter? Round to nearest the The circumference of acircular fountain is 32 . π . π . esn 1 Lesson DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-D;CA-D

Explain 2 Justifying the Area Formula

To find the area of a given circle, consider a regular polygon that is inscribed in the circle. As you increase the number of sides of the polygon, the area of the polygon gets closer to the area of the circle.

Inscribed Inscribed Inscribed pentagon hexagon octagon

Let circle O be a circle with center O and radius r. Inscribe a regular n–gon in circle O and draw radii from O to the vertices of the n-gon. _ _ Let AB be one side of the n–gon. Draw OM , the segment from O to the _ O midpoint of AB . _ _ We know that OM is perpendicular to AB because triangle AOM is congruent to 1 r r triangle BOM. h _ Let the length of OM be h. A x M x B Example 2 Justify the formula for the area of a circle.

A There are n triangles, all congruent to △AOB, that surround point O and fill the n-gon. _360° _180° Therefore, the measure of ∠AOB is n , and the measure of ∠1 is n .

_180° We know that x = r sin( n ) . Write a similar expression for h. B The area of AOB is _ 1 2x h xh r sin _180 ° h. △ 2( )( ) = = ( n ) Substitute your value for h to get area △AOB = . © Houghton Mifflin Harcourt Publishing Harcourt Company © Houghton Mifflin C There are n of these triangles, so the area of the n-gon is .

_180° _180°  Your expression for the area of the n–gon includes the factor n sin n cos n . What happens to this expression as n gets larger? ( ) ( )

Use your graphing calculator to do the following.

_180° _180° • Enter the expression x sin x cos x as Y 1 . • View a table for the function.( ) ( ) • Use arrow keys to scroll down. _180° _180° What happens to the value of x sin( x ) cos( x ) as x gets larger?  Look at the expression you wrote for the area of the n–gon. What happens to the value of this expression as n gets larger? .

Module 16 854 Lesson 1 CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes mustbemadethrough “File info”

oue 16 Module The 9-in.side of pizza the length the of is also the B The diameter of largest the circle is 3feet, or 36inches. The radius of circle the is 18inches. A Explain3 7. 6. Reflect 9. Your Turn 8. Reflect To nearest the square inch, area the of pizza the is So, area the is about 1,018square inches.

Example 3 Aslice of acircular pizza measures 9inches inlength. What is area the of entire the pizza? Arectangular piece of cloth by is 3ft What 6ft. is area the of largest the circle that answer to Example 3B?What additional information can you determine with fact? this How above the does argument formula the justify Why or why not? When the smallestthe amount of material to cover needed Use of surface the pool? the 3.14for A circular has swimming adiameter pool of 18feet. To nearest the square foot, what is Suppose slice the of pizza represents Use 3.14for cutcan from be cloth? the Round nearest the square inch. A A A A A A A

≈ ======1,017.9 in 324

Find area the indicated. π π π π n is very large, is very area the does of the

r ( r

2 2

π ) 2

2 π

≈ .

Applying theArea Formula _ 1 6

of your whole the affect this pizza. Does n- gon area the ever equal of circle? the 855 A

π r 2

? . of circle. the So, r

= π . . esn 1 Lesson © Houghton Mifflin Harcourt Publishing Company . Lesson Lesson 1 π • Online Homework • Hints and Help • Extra Practice . . π π . π 5 m 24 in. = =

A basketball rim has a radius of 9 inches. Find Find 9 inches. of A basketball rim a radius has the to the rim. Round of the circumference 3.14 for Use tenth. nearest Building’s Capitol the U.S. of diameter The its Find point. widest its at 96 feet is dome 3.14 for Use circumference. A drum has a diameter of 10 inches. Find the Find 10 inches. of A drum a diameter has 3.14 for the drum. of Use the top of area r d 5. 7. 11. 3. 9. 856 . π a circle? of the circumference for use the formula justifyand you do How . π 14.2 mm 16 ft 7 yd 9 cm

= = = =

Evaluate: Homework and Practice Homework Evaluate: The diameter of a circular swimming pool 12 swimming is a circular of diameter The 3.14 for Use circumference. its Find feet. d d r r Which inscribed figure has a perimeter closer to the circumference of a circle, a circle, of inscribed the circumference a perimeter to closer has figure Which Explain. 40 sides? with polygon a regular or 20 sides with polygon a regular Essential QuestionEssential Check-In If the radius of a circle is doubled, is the area doubled? Explain. doubled? the area is doubled, is a circle of the radius If

6. 4. Module Module 16 10. Find the area of each circle with the given radius or diameter. Use 3.14 for 3.14 for Use each diameter. with circle of the radius given or the area Find 8. nearest tenth. Use 3.14 for 3.14 for Use tenth. nearest 2. Find the circumference of each circle with the given radius or diameter. Round to the to Round each diameter. with circle the radius of given or the circumference Find 1. 11. 10. Elaborate Elaborate

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12. The circumference of a quarter is about 76 mm. What is the area? Round to the nearest tenth.

Algebra Find the area of the circle with the given circumference C. Use 3.14 for π.

13. C = 31.4 ft 14. C = 21.98 ft 15. C = 69.08 ft

16. A Ferris wheel has a diameter of 56 ft. How far will a rider travel during a 4-minute ride if the wheel rotates once every 20 seconds? Use _ 22 for . 7 π

• Image Credits:

17. A giant water lily pad is shaped like a circle with a diameter of up to 5 feet. Find the circumference and area of the pad. Round to the nearest tenth. © Houghton Mifflin Houghton © Company Harcourt Publishing ©Comstock/Getty(t) Images; ©slowfish/iStockPhoto.com (b)

Module 16 857 Lesson 1

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r w _ =

858 is the average the average is Which do you think would seat more people, a 4 ft by 6 ft a 4 ft by people, more seat think would do you Which

is the tree’s radius without bark and w and bark without radius the tree’s is r thickness of the tree’s rings. The circumference of a white oak tree a white of circumference The rings. the tree’s of thickness of thickness the average 0.5 in. thick, and is bark The 100 inches. is the enclosed by the area and age the tree’s 0.2 in. Estimate is a ring the ring. widest of circumference outer where where Critical Thinking A pizza parlor offers pizzas with diameters of 8 in., 10 in., and 12 in. Find the area the area Find 12 in. in., and 8 in., 10 of diameters with pizzas offers parlor A pizza $18 and $9, $12, cost the pizzas If tenth. the nearest to Round each size pizza. of buy? the which better is respectively, You can estimate a tree’s age in years by using the formula a the formula using by in years age a tree’s estimate can You rectangular table or a circular table with a diameter of 6 ft? How many people would would people many 6 ft? How of a diameter with table a circular or table rectangular reasoning. your Explain each table? at sit you

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21. Multi-Step A circular track for a model train has a diameter of 8.5 feet. The train moves around the track at a constant speed of 0.7 ft/s. a. To the nearest foot, how far does the train b. To the nearest minute, how long does it travel when it goes completely around the take the train to go completely around the track 10 times? track 10 times?

22. The Parthenon is a Greek temple dating to about 445 BCE. The temple features 46 Doric columns, which are roughly cylindrical. The circumference of each column at the base is about 5.65 meters. What is the approximate diameter of each column? Round to the nearest tenth.

23. Explain the Error A circle has a circumference of 2π in. Which calculation of the area is incorrect? Explain. A B The circumference is The circumference is 2π in., so the diameter 2π in., so the radius is 2 in. The area is is 1 in. The area is 2 2 2 2 A = π(2 ) = 4π in . A = π(1 ) = π in .

24. Write About It The center of each circle in the figure lies on the number line. Describe the relationship between the circumference of the largest circle and the circumferences of the four smaller circles.

0 10 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Adam Images Crowley/Photodisc/Getty 25. Find the diameter of a data storage disk with an area 113.1 cm 2.

Module 16 859 Lesson 1

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26. Which of the following ratios can be derived from the formula for the circumference of a circle? Select all that apply. A. _ C d B. _ C 2r C. _C π _C D. 2π E. _C 2c

27. A meteorologist measured the eyes of hurricanes to be 15 to 20 miles in diameter during one season. What is the range of areas of the land underneath the eyes of the hurricanes?

28. A circle with a 6 in. diameter is stamped out of a rectangular piece of metal as shown. Find the area of the remaining piece of metal. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Stocktrek ©Stocktrek Credits: • Image Company Publishing Harcourt Mifflin © Houghton Use 3.14 for π.

14 in.

8 in.

H.O.T. Focus on Higher Order Thinking

29. Critique Reasoning A standard bicycle wheel has a diameter of 26 inches. A Images Images, Inc./Getty student claims that during a one-mile bike ride the wheel makes more than 1000 complete revolutions. Do you agree or disagree? Explain. (Hint: 1 mile = 5280 feet)

Module 16 860 Lesson 1

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30. Algebra A graphic artist created a company logo from two tangent circles whose diameters are in the ratio 3:2. What percent of the total logo is the area of the region outside of the smaller circle?

_ 31. Communicate Mathematical Ideas In the figure, AB is a diameter of _ _ circle C, D is the midpoint of AC , and E is the midpoint of AD . How does the circumference of circle E compare to the circumference of circle C? E D C Explain. A B

32. Critical Thinking Evelyn divides the circle and rearranges the pieces to make a shape that resembles a parallelogram.

πr

She then divides the circle into more pieces and rearranges them into a shape that resembles a parallelogram. She thinks that the area of the new parallelogram will be closer to the area of a circle. Is she correct? Explain. © Houghton Mifflin Houghton © Company Harcourt Publishing

= = 10 5 2.5 + that day that +

10 (in kilometers) Total distance walked walked distance Total

862 5 10 2.5 (in kilometers) from the day before the day from Increase in distance walked walked distance in Increase Even though you can’t cut a circle into millions of tiny wedges to calculate calculate to wedges tiny of millions into a circle cut can’t you though Even

limit. 5 6 4 3 1 2 Day Complete the table for Mac’s first 6 days of training. of 6 days first Mac’s for the table Complete where he walks 20 kilometers a day. walks he 20 kilometers where reach ever Mac Will plan. training his continues Mac Suppose goal? his Describe your results in relation to Mac’s plan to reach a level a level reach to plan Mac’s to in relation Describe results your

1. 3. 2.

10 kilometers the first day of training and increase the distance he walks each day until he he until walks he day each the distance increase and training of the day first 10 kilometers his goal.reaches spotting a pattern. You can apply this method in many ways, some of them unexpected. of some ways, this method in many apply can You a pattern. spotting to up leading the in weeks decided has that He a race. for training is He walker. a race is Mac walk to is plan His a day. walking is 20 kilometers he where a level to up work he’ll the race, reassembled wedges becomes. In the branch of mathematics called calculus, this process is is called calculus, this process mathematics of the branch In becomes. wedges reassembled called a finding by that, before be to long going is the see area can what you the circle, of the actual area In the lesson, you saw that the more wedges into which you divide a circle, the more the more a circle, divide you which into wedges the more that saw you the lesson, In the of the area a circle of the area the to closer and a triangle, resembles wedge each closely Lesson Lesson Task Performance Module 16 Module

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