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10.11 Length of an arc (a) Find the length of a 36 arc in a circle whose circumference is 45p. (b) Find the radius of a circle if a 40 arc has a length of 4p.
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(a) Here n (b) Here l 36 and C 4p and n 2pr 45p. Then l n 2pr 360 36 9 45p p. 360 2 40 2pr, and r 360
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40 . Then l
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n 2pr yields 4p 360
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10.12 Area of a sector (a) Find the area K of a 300 sector of a circle whose radius is 12. (b) Find the measure of the central angle of a sector whose area is 6p if the area of the circle is 9p. (c) Find the radius of a circle if an arc of length 2p has a sector of area 10p.
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CHAPTER 10 Regular Polygons and the Circle
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(a) n (b) (c) 300 and r 12. Then K 6p n , so 360 9p n pr2 360 300 144p 360 120p.
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Area of sector Area of circle Length of arc Circumference
n , and n 360
240. Thus, the central angle measures 240 . 10.
2p area of sector , so area of circle 2pr
10p and r pr2
10.13 Area of a segment of a circle (a) Find the area of a segment if its central angle measures 60 and the radius of the circle is 12. (b) Find the area of a segment if its central angle measures 90 and the radius of the circle is 8. (c) Find each segment formed by an inscribed equilateral triangle if the radius of the circle is 8.
Solutions
See Fig. 10-11. (a) n 60 and r 12. Then area of sector OAB
1 2 4 s 23
Also, area of equilateral ^OAB Hence, area of segment ACB (b) n 90 and r
n pr2 360 1 4 (144) 23
60 144p 360 36 23.
24p.
36 23. n pr2 360 90 64p 360 16p.
8. Then area of sector OAB
1 2 bh 1 2 (8)(8)
Also, area of rt. ^OAB
32. 48 23.
Hence, area of segment ACB
1 (c) R 8. Since s R 23 8 23, the area of ^ABC is 4s2 23 Also, area of circle O pR2 64p. 1 Hence, area of segment BDC 3(64p 48 23).
Fig. 10-11
10.14 Area of a segment formed by an inscribed regular polygon Find the area of each segment formed by an inscribed regular polygon of 12 sides (dodecagon) if the radius of the circle is 12. (See Fig. 10-12.)
Fig. 10-12
CHAPTER 10 Regular Polygons and the Circle
Solution
n 30 pr2 144p 12p. 360 360 To find the area ^OAB, we draw altitude AD to base OB. Since m/AOB Area of sector OAB Then the area of ^OAB is
1 2 bh 1 2 (12)
30 , h
1 2r
(6) 36.
Hence, the area of segment ACB is 12p
10.7 Areas of Combination Figures
The areas of combination figures like that in Fig. 10-13 may be found by determining individual areas and then adding or substracting as required. Thus, the shaded area in the figure equals the sum of the aeas of the square and the semicircle: A 82 1(16p) 64 8p. 2
Fig. 10-13
SOLVED PROBLEMS
10.15 Finding areas of combination figures Find the shaded area in each part of Fig. 10-14. In (a), circles A, B, and C are tangent externally and each has radius 3. In (b), each arc is part of a circle of radius 9.
Fig. 10-14
Solutions
(a) Area of ^ABC Shaded area (b) Area of square Shaded area
1 2 4 s 23
9 23 182 324
1 2 (6 ) 23 9 23. Area of sector I 4 15 3 A 2 pR 9 23 45p. 2 n (pr2) 360
n (pr2) 360
300 (9p) 360
15 p. 2
324. Area of sector I 4 A 4 pB
90 (81p) 360
81 p.1 4
81p.
CHAPTER 10 Regular Polygons and the Circle
SUPPLEMENTARY PROBLEMS
10.1. In a regular polygon, find (a) The perimeter if the length of a side is 8 and the number of sides is 25 (b) The perimeter if the length of a side is 2.45 and the number of sides is 10 (c) The perimeter if the length of a side is 42 and the number of sides is 24 3 (d) The number of sides if the perimeter is 325 and the length of a side is 25 (e) The number of sides if the perimeter is 27 23 and the length of a side is 3 23 (f) The length of a side if the number of sides is 30 and the perimeter is 100 (g) The length of a side if the perimeter is 67.5 and the number of sides is 15 10.2. In a regular polygon, find (a) The length of the apothem if the diameter of an inscribed circle is 25 (b) The length of the apothem if the radius of the inscribed circle is 23.47 (c) The radius of the inscribed circle if the length of the apothem is 7 23 (d) The radius of the regular polygon if the diameter of the circumscribed circle is 37 (e) The radius of the circumscribed circle if the radius of the regular polygon is 3 22 10.3. In a regular polygon of 15 sides, find the measure of (a) the central angle; (b) the exterior angle; (c) the interior angle. (10.1) 10.4. If an exterior angle of a regular polygon measures 40 , find (a) the measure of the central angle; (b) the number of sides; (c) the measure of the interior angle. (10.1) 10.5. If an interior angle of a regular polygon measures 165 , find (a) the measure of the exterior angle; (b) the measure of the central angle; (c) the number of sides. (10.1) 10.6. If a central angle of a regular polygon measures 5 , find (a) the measure of the exterior angle; (b) the number of sides; (c) the measure of the interior angle. (10.1) 10.7. Name the regular polygon whose (a) Central angle measures 45 (b) Central angle measures 60 (c) Exterior angle measures 120 10.8. Prove each of the following: (a) The diagonals of a regular pentagon are congruent. (b) A diagonal of a regular pentagon forms an isosceles trapezoid with three of its sides. (c) If two diagonals of a regular pentagon intersect, the longer segment of each diagonal is congruent to a side of the regular pentagon. 10.9. In a regular hexagon, find (a) The length of a side if its radius is 9 (b) The perimeter if its radius is 5 (10.3) (d) Exterior angle measures 36 (e) Interior angle is congruent to its central angle (f) Interior angle measures 150 (10.2) (10.1) (10.1) (10.1)
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