DUGV - Seattle, WA

CONSTRUCTION Find the area of each circle.Round to the nearest tenth.

1. Refer to the figure on page 745. The circle in thephoto has a radius of 21 yards.

SOLUTION:

ANSWER:

2. Refer to the figure on page 745. The circle in thephoto has a diameter of 0.4 kilometers.

SOLUTION:

ANSWER:

Find the indicated measure. Round to thenearest tenth.

3. Find the diameter of a circle with an area of 74square millimeters.

SOLUTION:

ANSWER:

9.7 mm

4. The area of circle is 88 square inches. Find theradius.

SOLUTION:

ANSWER:

5.3 in.

eSolutions Manual - Powered by Cognero Page 1

10-3 Areas of Circles and Sectors

Find the area of each shaded sector. Round tothe nearest tenth.

5.

SOLUTION:

The ratio of the area A of a sector to the area of the

whole circle, πr2, is equal to the ratio of the degreemeasure of the intercepted arc x to 360.

ANSWER:

6.

SOLUTION:


whole circle, πr2, is equal to the ratio of the degreemeasure of the intercepted arc x to 360.

ANSWER:

7. BAKING Chelsea is baking pies for a fundraiser ather school. She divides each 9-inch pie into 6 equalslices.a. What is the area, in square inches, for each slice ofpie?b. If each slice costs $0.25 to make and she sells 8pies at $1.25 for each slice, how much money willshe raise?

SOLUTION:

a. Since the pie is equally divided into 6 slices, eachslice will have an arc measure of 360 ÷ 6 or 60.

b. The manufacturing cost for each slice is $0.25 andshe sells it for $1.25. So, she makes a profit of $1from each slice of 8 pies. There are 6 slices in eachpie. So, the total profit is 8(6)(1) = 48. Therefore, shewill raise an amount of $48.

ANSWER:

a. b. $48



MODELING Find the area of each circle.Round to the nearest tenth.

8. Refer to the figure on page 746.

SOLUTION:

ANSWER:


SOLUTION:

ANSWER:


SOLUTION:

ANSWER:


SOLUTION:

ANSWER:




SOLUTION:

ANSWER:


SOLUTION:

ANSWER:

Find the indicated measure. Round to thenearest tenth, if necessary.

14. The area of a circle is 68 square centimeters. Findthe diameter.

SOLUTION:

ANSWER:

9.3 cm

15. Find the diameter of a circle with an area of 94square millimeters.

SOLUTION:

ANSWER:

10.9 mm



16. The area of circle is 112 square inches. Find theradius.

SOLUTION:

ANSWER:

6 in.

17. Find the radius of a circle with an area of 206 squarefeet.

SOLUTION:

The area A of a circle is equal to π times the squareof the radius r.

The radius of the circle is about 8.1 in.

ANSWER:

8.1 ft

Find the area of each shaded sector. Round tothe nearest tenth, if necessary.

18.

SOLUTION:

ANSWER:

19.

SOLUTION:

ANSWER:



20.

SOLUTION:

ANSWER:

21.

SOLUTION:

ANSWER:

22.

SOLUTION:

ANSWER:

23.

SOLUTION:

ANSWER:

24. MUSIC The music preferences of students atThomas Jefferson High are shown in the circlegraph. Find the area of each sector and the degreemeasure of each intercepted arc if the radius of thecircle is 1 unit.



SOLUTION: Multiply each percentage by 360° to find the measureof the corresponding arc. The area of the circle is π units. Multiply eachpercentage by this to find the area of eachcorresponding sector. rap:

rock & roll:

alternative:

country:

classical:

ANSWER:

rap: 172.8°, 1.51 units2;

rock & roll: 93.6°, 0.82 units2;

alternative: 50.4°, 0.44 units2;

country: 36°, 0.31 units2;

classical: 7.2°, 0.06 units2

25. JEWELRY A jeweler makes a pair of earrings bycutting two 50° sectors from a silver disk.a. Find the area of each sector.b. If the weight of the silver disk is 2.3 grams, howmany milligrams does the silver wedge for eachearring weigh?

SOLUTION:

a.

b. of the disc has been removed to make each

earring. So, the weight of each earring is

ANSWER:

a. b. 319.4 mg

26. PROM Students voted on their favorite promtheme.

a. Create a circle graph with a diameter of 2 inchesto represent these data.b. Find the area of each theme’s sector in yourgraph. Round to the nearest hundredth of an inch.

SOLUTION: eSolutions Manual - Powered by Cognero Page 7


a. Multiply each percentage by 360 to find thedegree measure of each sector.

Use these measures to create the sectors of thecircle.

b. We are given the percentages, so multiply the areaof the circle, π, by each percentage.

ANSWER:

a.

b. An Evening of Stars: ; Mardi Gras:

; Springtime in Paris: ; Night in

Times Square: ; Undecided:

SENSE-MAKING The area A of each shadedregion is given. Find x.

27.

SOLUTION:


whole circle, πr2, is equal to the ratio of the degreemeasure of the intercepted arc x to 360. So, the area

A of a sector is given by

The value of x, which is the diameter of the circle, isabout 13 cm.

ANSWER:

13



28.

SOLUTION:




ANSWER:

55

29.

SOLUTION:




x in the diagram is the radius, r.

ANSWER:

9.8

30. MULTI-STEP Luna is organizing a banquet for theHonor Society, and she needs 13 tablecloths for theround tables in the hall. The area of each table isapproximately 29.27 square feet. She can renttablecloths for $16 each or she can make themherself. Her local fabric store carries three differentbolts of suitable fabric. The standard bolt is 60inches wide and 100 yards long and costs $75. Thewide bolt is 81 inches wide, 25 yards long, andcosts $125. The extra-wide bolt is 90 inches wide,25 yards long, and costs $150. Each tableclothshould cover the table with 9 inches of overhang.a. How can Luna minimize the cost of thetablecloths?b. Explain your reasoning.



c. What assumptions did you make?

SOLUTION:

a-b. The area of the table is about 29.27 . Usingthe formula for the area of a circle, , findthe radius and diameter for the tablecloth.

The radius is about 3 ft, so the diameter is about 6ft. She wants the fabric to extend 9 inches over theedge of the table, so add 18 inches to the diameterfor a total of 6(12) + 18 or 90 inches. The only boltof fabric that could be used is the widest bolt ( 81 x25).Cut the fabric into 90-in squares and then cutcircles. Converting the width of the bolt into inches,you get

. Divide this by90 inches needed for one tablecloth and Luna canmake 10 tablecloths from a bolt at a cost of $150.Each tablecloth would cost $15. However, shewould still need to rent 3 tablecloths to cover all ofthe tables for a total cost of $198.c. Assumptions made were that there were no othercosts associated with making her own tablecloths;she only had to buy the fabric. Also, it was assumedthat it didn’t matter that the tablecloths didn’t match.

ANSWER:

a. She should rent 3 tablecloths and make 10tablecloths from the 90″ wide bolt.b. The area of the table is about 29.27 . Usingthe formula for the area of a circle, , we canfind the radius and diameter for the tablecloth. Theradius is about 3 ft, so the diameter is about 6 ft.She wants the fabric to extend 9 inches over theedge of the table, so add 18 inches to the diameterfor a total of 6(12) + 18 or 90 inches. The only bolt

of fabric that could be used is the widest bolt. Cutthe fabric into 90-in squares and then cut circles.25(3)(12) ÷ 90 = 10, so Luna can make 10tablecloths from a bolt at a cost of $150. Eachtablecloth would cost $15. However, she would stillneed to rent 3 tablecloths to cover all of the tablesfor a total cost of $198.c. I assumed that there were no other costsassociated with making her own tablecloths; sheonly had to buy the fabric. I also assumed that itdidn’t matter that the tablecloths didn’t match.

31. TREES The age of a living tree can be determinedby multiplying the diameter of the tree by its growthfactor, or rate of growth. The Coast Live Oak is thelargest tree in Texas.a. What is the diameter of a live oak tree with acircumference of 36 feet?b. If the growth factor of the live oak tree is 130,what is the age of the tree?

SOLUTION:

a. Using the given circumference, find the diameterof the tree.

b. Multiply the growth factor by the diameter to findthe age.Age = (130)(11.5) = 1495 yr

ANSWER:

a. 11.5 ftb. 1495 yr



Find the area of the shaded region. Round tothe nearest tenth.

32.

SOLUTION:

The area of the shaded region is the differencebetween the area of the square and that of thecircle. The length of each side of the square is 18 ft and theradius of the circle is 9 ft.

ANSWER:

33.

SOLUTION:

The area of the shaded region is the differencebetween the area of the circle and that of thetriangle.

Draw a radius from to the bottom vertex of thetriangle. This is an isosceles triangle where the legsare the radius. Using Pythagorean Theorem to find r.

The height of the triangle is the radius of the circle: 5cm.

The area of the shaded region is about 53.5 m2.

ANSWER:

53.5 m2



34.

SOLUTION:

The area of the shaded region is the differencebetween the area of the larger circle and the sum ofthe areas of the smaller circles. The diameter of the larger circle is 14 mm, so theradius is 7 mm. The two smaller circles are congruentto each other and the sum of their diameters is 14mm. So, the radius of each of the congruent smallcircles is 3.5 mm.

ANSWER:

35.

SOLUTION:

The two smaller circles are congruent to each otherand the sum of their diameters is 10 cm, so the radiusof each of the circles is 2.5 cm.

ANSWER:

36.

SOLUTION:

The angles of the sectors are each a linear pair withthe 130° angle.

ANSWER:

37.

SOLUTION:

The sum of the central angles of the shaded sectorsis 360 – 3(45) = 225.

ANSWER:



38. COORDINATE GEOMETRY What is the area ofsector ABC shown on the graph?

SOLUTION:

ANSWER:

28.3 square units

39. ALGEBRA The figure shown below is a sector of acircle. If the perimeter of the figure is 22 millimeters,find its area in square millimeters.

SOLUTION:

The length of the arc is 22 – (6 + 6) = 10.

Use the measure of the central angle to find the areaof the sector.

ANSWER:



Find the area of each shaded region.

40.

SOLUTION:

The radius of the larger circle is 4.5 cm and theradius of the small circle is 4.5 ÷ 2 = 2.25 cm.

ANSWER:

41.

SOLUTION:

The larger circle has a radius of 6 in. The three smaller circles are congruent and the sumof their diameters is 12 in. So, each has a radius of 2in. The area of the shaded region is half of the largecircle minus half of one of the small circles. Note thatthe shaded half circle offsets one of the unshadedhalf circles.

ANSWER:

42.

SOLUTION:

The area of the shaded region is the differencebetween the area covered by the minor arc and thearea of the triangle. The central angle of the minor arc is 360 – 240 =120.

The central angle is 60°, so the triangle is equilateral.



Use 36-60-90 triangles to find the height.

Find the area of the triangle.

ANSWER:

43. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate segments of circles. Asegment of a circle is the region bounded by an arcand a chord.a. ALGEBRAIC Write an equation for the area Aof a segment of a circle with a radius r and a centralangle of x°. (Hint: Use trigonometry to find the baseand height of the triangle.)

b. TABULAR Calculate and record in a table tenvalues of A for x-values ranging from 10 to 90 if r is12 inches. Round to the nearest tenth.c. GRAPHICAL Graph the data from your tablewith the x-values on the horizontal axis and the A-values on the vertical axis.d. ANALYTICAL Use your graph to predict the

value of A when x is 63. Then use the formula yougenerated in part a to calculate the value of A when xis 63. How do the values compare?

SOLUTION:

a. Draw a perpendicular from the center to the chordto get two congruent triangles whose hypotenuse is runits long. Let the height of the triangle be h and the length ofthe chord, which is a base of the triangle be .

Use trigonometry to find l and h in terms of r and x.

Now find the area of the triangle.

Lastly, find the area of the segment.



b. If r = 12, then the new formula is:

Enter this formula into Y1 of your calculator. Theselect the table function and set the range for 10 to90 by 10.

c. Plot the values. One other option would be to enter10 through 90 by 10 in L1 and enter the formula forL2, replacing x with L1. Then, you can selectSTATPLOT L1, L2.

d. Sample answer: From the graph, it looks like thearea would be about 15.5 when x is 63°.

Therefore, the area of the segment is about 15.0when x is 63°.

ANSWER:

a.

b.



c.

d. Sample answer: From the graph, it looks like thearea would be about 15.5 when x is 63°. Using theformula, the area is 15.0 when x is 63°. The valuesare very close because I used the formula to createthe graph.

44. ERROR ANALYSIS Kristen and Chase want tofind the area of the shaded region in the circle shown.Is either of them correct? Explain your reasoning.

SOLUTION:

The diameter of the circle is given to be 8 in., so theradius is 4 in. Therefore, Chase is correct. Kristenused the diameter in the area formula instead of theradius.

ANSWER:

Chase; sample answer: Kristen used the diameter inthe area formula instead of the radius.



45. CHALLENGE Find the area of the shaded region.Round to the nearest tenth.

SOLUTION:

The radius of the larger circle is 17.5 cm and that ofthe smaller circle is 7 cm. The measure of the central angle of the shadedregion is 360 – 160 = 200.

ANSWER:

449.0 cm2

46. CONSTRUCT ARGUMENTS Refer to Exercise43. Is the area of a sector of a circle sometimes,always, or never greater than the area of itscorresponding segment?

SOLUTION:

In most cases, the area of the sector (as designatedby the blue region) is greater than the area of thesegment (as designated by the red region) for thesame central angle. The area of the segment iscontained within the area of the sector. In fact, tocalculate the area of the segment, you need tosubtract the area of the triangle determined by thecentral angle and the chord from the area of thesector.

However, if the central angle and the chord bothintercept a semicircle, the area of the sector and thearea of the segment (as designated by the brownregion) are equal.

Therefore, the statement is sometimes true.

ANSWER:

Sometimes; when the arc is a semicircle, the areasare the same.

47. WRITING IN MATH Describe two methods youcould use to find the area of the shaded region of thecircle. Which method do you think is more efficient?Explain your reasoning.



SOLUTION:

Method 1:You can find the shaded area of the circle bysubtracting x from 360° and using the resultingmeasure in the formula for the area of a sector. Let x = 120 and r = 10.360 – 120 = 240

Method 2: You could find the shaded area by findingthe area of the entire circle, finding the area of theun-shaded sector using the formula for the area of asector, and subtracting the area of the un-shadedsector from the area of the entire circle. Let x = 120 and r = 10.

The method in which you find the ratio of the area ofa sector to the area of the whole circle is moreefficient. It requires fewer steps, is faster, and thereis a lower probability for error.

ANSWER:

Sample answer: You can find the shaded area of thecircle by subtracting x from 360° and using theresulting measure in the formula for the area of asector. You could also find the shaded area by findingthe area of the entire circle, finding the area of theun-shaded sector using the formula for the area of asector, and subtracting the area of the un-shadedsector from the area of the entire circle. The methodin which you find the ratio of the area of a sector tothe area of the whole circle is more efficient. Itrequires fewer steps, is faster, and there is a lowerprobability for error.

48. CHALLENGE Derive the formula for the area of asector of a circle using the formula for arc length.

SOLUTION: Think of how the arc length and the area of a sectorare related to the circle as a whole. For instance, halfof a circle will have half of the arc length and half ofthe area of the whole circle. A quarter of a circle willhave a quarter of the arc length and a quarter of thearea. We can express each of these casesmathematically as follows: Half circle:

Quarter circle:

From this we should deduce that the ratio of the areaof a sector to the area of the circle should be thesame ratio as the arc length divided by thecircumference. Next, we express this mathematicallyand using known formulas derive the area for asector. Let A represent the area of the sector.



ANSWER: The ratio of the area of a sector to the area of awhole circle is equal to the ratio of the correspondingarc length to the circumference of the circle. Let Arepresent the area of the sector.

49. WRITING IN MATH If the radius of a circle

doubles, will the measure of a sector of that circledouble? Will it double if the arc measure of thatsector doubles?

SOLUTION:

If the radius of the circle doubles, the area will notdouble.

If the radius of the circle doubles, the area will befour times as great. 36π = 4(9π) If the arc length of a sector is doubled, the area ofthe sector is doubled.

Since the arc length is not raised to a power, if thearc length is doubled, the area would also be twice aslarge.

ANSWER:

Sample answer: If the radius of the circle doubles,the area will not double. If the radius of the circledoubles, the area will be four times as great. Since



the radius is squared, if you multiply the radius by 2,

you multiply the area by , or 4. If the arc length ofa sector is doubled, the area of the sector is doubled.Since the arc length is not raised to a power, if thearc length is doubled, the area would also be twice aslarge.

50. Visitors at a school carnival have a change to toss abean onto a circular tabletop that is divided into equalsectors, as shown.

Visitors win a prize if the bean lands in the shadedsector. What is the area of this sector in squareinches? Round to the nearest tenth.

SOLUTION: The circle is divided into 12 equal sections. Substituteinto area formula and divide by 12.

The area is about 84.8 square inches.

ANSWER: 84.8

51. A lawn sprinkler sprays water 25 feet and movesback and forth through an angle of 150°.

Which of the following is the best estimate of thearea of the lawn that gets watered?

A 65

B 818

C 1963

D 4712

SOLUTION: Use the Area of a Sector formula to find the area ofthe lawn that gets watered:

The correct choice is B.

ANSWER: B



52. A sector of a circle has an intercepted arc thatmeasures 120°. The area of the sector is 155.8square centimeters. What is the radius of the circle incentimeters? Round to the nearest tenth.

SOLUTION: Use the Area of a Sector formula to solve for theradius of the circle:

ANSWER: 12.2

53. One pizza with radius 9 inches is cut into 8 congruentsectors. Another pizza with the same radius is cutinto 10 congruent sectors. How much more pizza, insquare inches, is in a slice from the pizza cut into 8sectors?

A 6.4B 25.4C 31.8D 57.2

SOLUTION: Find the difference between one-eighth of a circleand one-tenth of a circle with a radius of 9 inches.

The larger slices are about 6.4 square inches larger.The correct choice is A.

ANSWER: A



54. Which expression represents the area of the shadedsector in square meters?sector in square meters?

A

B C 2πD 4π

SOLUTION: First, find the radius of the circle. Since the shadedtriangle is a right isosceles triangle, then it is a 45-45-90 special right triangle. Find the legs by dividing the

hypotenuse by :

Now, use the Area of a Sector formula:

The correct choice is C.

ANSWER: C

55. In ⊙C, a sector has an area of 24π square inches.The radius of ⊙C is 12 inches. What is the measure,in degrees, of the arc that is intercepted by thesector? A 360B 60πC 60D 180

SOLUTION: Use the Area of the Sector of a Circle formula:

The correct choice is C.

ANSWER: C



56. A circular pie has a diameter of 8 inches and is cutinto 6 congruent slices. What is the area of one sliceof pie?

A 6 square inchesB 8.4 square inchesC 21.6 square inchesD 33.5 square inches

SOLUTION: One slice of pie is one sixth of the pie. Multiply thearea of the pie times one-sixth.

The area of one slice of pie is about 33.5 squareinches. The correct choice is D.

ANSWER: D

57. MULTI-STEP A regular hexagon, inscribed in acircle, is divided into 6 congruent triangles. Theperimeter of the hexagon is 48 inches.

a. What is the radius of the circle?

b. Find the area of each of the 6 sectors of the circlethat have sides that coincide with sides of thecongruent triangles. Round to the nearest tenth.

c. What is the area of one of the triangles? Round tothe nearest tenth.

d. How much greater is the sector area than that ofone of the triangles? Round to the nearest tenth.

SOLUTION: a. The radius of the circle is equal to one side of thehexagon. Since the hexagon is regular with aperimeter of 48 inches, each side is 8 inches, so theradius is 8 inches.

b. The area of each sector is one-sixth of the circle.

The area of each sector is about 33.5 square inches.

c. The area of each triangle is one half base times

height. The base is 8 inches and the height is inches, since each triangle is equilateral.

The area of each triangle is about 27.7 square inches.

d. Each sector is 33.5 - 27.7 = 5.8 square incheslarger than the triangle inside it.

ANSWER: a. 8 inches

b. 33.5 square inches

c. 27.7 square inches

d. 5.8 square inches



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