Circular measure

1. Convert 135 degrees to radians, giving an exact answer.

Answer A: 3 pi/4

135(pi/180) = (135/180)pi = 3 pi/4 radians. The conversion factor changes the unit, not the angle.

2. Convert 5 pi/9 radians to degrees.

Answer B: 100 degrees

(5 pi/9)(180/pi) = 5(20) = 100 degrees. This lies between pi/2 and pi, as expected.

3. Which statement defines an angle of one radian at the centre of a circle?

Answer C: It subtends an arc whose length equals the radius.

One radian is the central angle for which s/r = 1, so the subtended arc length s equals the radius r.

4. A circle has radius 8 cm. A minor arc of length 14 cm subtends angle theta at the centre. Find theta in radians.

Answer D: 7/4

theta = s/r = 14/8 = 7/4 radians. No radii are added because the 14 cm is explicitly the curved arc.

5. A minor arc in a circle of radius 12 cm subtends 0.65 radians at the centre. Find the arc length.

Answer A: 7.8 cm

s = r theta = 12(0.65) = 7.8 cm. This is the curved arc alone, not a sector perimeter.

6. A sector has radius 5 cm and angle 2.4 radians. Its boundary is the minor arc plus both radii. Find its perimeter.

Answer B: 22 cm

The arc is 5(2.4) = 12 cm. The full sector perimeter is 12 + 5 + 5 = 22 cm.

7. Two radii of a circle of radius 3 cm form a minor angle of 1.1 radians. Find the length of the major arc joining their endpoints.

Answer C: 3(2 pi - 1.1) cm

The major angle is 2 pi - 1.1, so its arc length is r theta = 3(2 pi - 1.1) cm.

8. A semicircle has radius 7 cm. What is the length of its curved arc only?

Answer D: 7 pi cm

The curved arc is r theta = 7(pi) = 7 pi cm. A diameter would be added only for the perimeter of the semicircular region.

9. A sector has radius 8 cm and central angle 0.75 radians. Find its area.

Answer A: 24 cm^2

A = (1/2)(8^2)(0.75) = 32(0.75) = 24 cm^2. The angle is already in radians, so no conversion is required.

10. A sector of radius 6 cm has area 45 cm^2. Find its central angle in radians.

Answer B: 2.5

theta = 2(45)/6^2 = 90/36 = 2.5 radians. The result is below pi, so it can describe a minor sector.

11. An arc has length 15 cm and subtends 1.25 radians at the centre. Find the circle radius.

Answer C: 12 cm

r = s/theta = 15/1.25 = 12 cm. Substitution checks that 12(1.25) = 15.

12. Which operation correctly converts an angle x degrees into radians?

Answer D: Multiply x by pi/180.

Since each degree equals pi/180 radians, x degrees equals x(pi/180) radians.

13. In a circle of radius 6 cm, chord AB subtends the minor angle pi/2 at centre O. Find the area of the minor segment bounded by chord AB and the minor arc AB.

Answer A: 9 pi - 18 cm^2

Sector area is (1/2)(36)(pi/2) = 9 pi. Triangle area is (1/2)(36)sin(pi/2) = 18, so the segment is 9 pi - 18 cm^2.

14. For 0 < theta < pi, which expression gives the minor segment area when radius r and minor central angle theta radians are known?

Answer B: sector area - triangle area

The minor segment is the portion of the minor sector outside triangle AOB, so its area is sector area minus triangle area.

15. A circle has radius 4 cm. A stated minor angle is pi/3. Find the area of the major sector determined by the same two radii.

Answer C: 40 pi/3 cm^2

The major angle is 5 pi/3. Thus area = (1/2)(4^2)(5 pi/3) = 40 pi/3 cm^2.

16. Two radii OA and OB each have length 10 cm and their included angle AOB is 0.6 radians. Find the area of triangle AOB.

Answer D: 50 sin(0.6) cm^2

Triangle area = (1/2)(10)(10)sin(0.6) = 50 sin(0.6) cm^2. The sine is evaluated in radians.

17. Two concentric circles have radii 5 cm and 3 cm. Two common rays form an angle of 2 radians. Find the area between the circles and between the rays.

Answer A: 16 cm^2

The annular-sector area is (1/2)(5^2 - 3^2)(2) = (1/2)(16)(2) = 16 cm^2.

18. An annular sector has outer radius 6 cm, inner radius 2 cm and angle pi/2. Its boundary contains both arcs and two radial connectors. Find its perimeter.

Answer B: 4 pi + 8 cm

The arcs total 6(pi/2) + 2(pi/2) = 4 pi. The connectors total 2(6 - 2) = 8, giving 4 pi + 8 cm.

19. In a circle of radius 9 cm, chord AB subtends a minor angle of 1 radian at O. Find the perimeter of the minor segment bounded by chord AB and the minor arc AB.

Answer C: 9 + 18 sin(0.5) cm

The arc is 9(1) = 9 cm and the chord is 2(9)sin(0.5). The segment perimeter is 9 + 18 sin(0.5) cm.

20. A minor sector has area 32 cm^2, and the triangle formed by its two radii and chord has area 12 cm^2. Find the minor segment area.

Answer D: 20 cm^2

Minor segment area = sector area - triangle area = 32 - 12 = 20 cm^2, because the central triangle lies inside the minor sector.

21. A sector has radius 8 cm and curved arc length 5 cm. Find its area without first rounding the angle.

Answer A: 20 cm^2

Since theta = s/r, A = (1/2)r^2(s/r) = (1/2)rs = (1/2)(8)(5) = 20 cm^2.

22. A sector has radius 9 cm. Its complete perimeter, consisting of one minor arc and two radii, is 30 cm. Find its angle in radians.

Answer B: 4/3

The arc length is 30 - 18 = 12 cm. Hence theta = 12/9 = 4/3 radians; using 30/9 would wrongly treat the full perimeter as an arc.

23. A circle has radius 5 cm, and chord AB subtends the minor angle 1.2 radians. Which expression is the area of the major segment bounded by chord AB and the major arc AB?

Answer C: 25 pi - 15 + 12.5 sin(1.2)

Minor segment area is 12.5(1.2) - 12.5sin(1.2) = 15 - 12.5sin(1.2). Subtracting it from 25 pi gives 25 pi - 15 + 12.5sin(1.2).

24. In a circle of radius 7 cm, points A and B subtend the minor angle 2 radians at the centre. Find the straight chord length AB.

Answer D: 14 sin(1) cm

Half the chord is 7 sin(2/2) = 7 sin(1), so AB = 14 sin(1) cm. This is a straight chord, not the arc length 14 cm.