Circular measure

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

1. 3 pi/4
2. 2 pi/3
3. 4 pi/3
4. 3 pi/2

2. Convert 5 pi/9 radians to degrees.

1. 80 degrees
2. 100 degrees
3. 120 degrees
4. 225 degrees

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

1. It subtends a chord equal to the diameter.
2. It subtends a semicircular arc.
3. It subtends an arc whose length equals the radius.
4. It subtends an arc whose length equals the circumference.

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

1. 4/7
2. 7/8
3. 8/7
4. 7/4

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

1. 7.8 cm
2. 8.45 cm
3. 12.65 cm
4. 18.46 cm

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

1. 12 cm
2. 22 cm
3. 24 cm
4. 34 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.

1. 3.3 cm
2. 3(2 pi + 1.1) cm
3. 3(2 pi - 1.1) cm
4. 6 pi - 1.1 cm

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

1. 7 pi/2 cm
2. 14 cm
3. 14 pi cm
4. 7 pi cm

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

1. 24 cm^2
2. 32 cm^2
3. 48 cm^2
4. 64 cm^2

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

1. 1.25
2. 2.5
3. 5
4. 15

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

1. 8 cm
2. 10 cm
3. 12 cm
4. 18.75 cm

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

1. Multiply x by 180/pi.
2. Add pi/180 to x.
3. Divide x by 2 pi.
4. Multiply x by pi/180.

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.

1. 9 pi - 18 cm^2
2. 9 pi cm^2
3. 18 cm^2
4. 18 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?

1. sector area + triangle area
2. sector area - triangle area
3. circle area - sector area
4. triangle area - sector 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.

1. 8 pi/3 cm^2
2. 16 pi/3 cm^2
3. 40 pi/3 cm^2
4. 16 pi 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.

1. 10 sin(0.6) cm^2
2. 25 sin(0.6) cm^2
3. 100 sin(0.6) cm^2
4. 50 sin(0.6) cm^2

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.

1. 16 cm^2
2. 20 cm^2
3. 32 cm^2
4. 34 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.

1. 4 pi cm
2. 4 pi + 8 cm
3. 8 pi + 4 cm
4. 16 pi 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.

1. 9 + 9 sin(1) cm
2. 18 + 9 sin(0.5) cm
3. 9 + 18 sin(0.5) cm
4. 18 + 18 sin(1) 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.

1. 12 cm^2
2. 16 cm^2
3. 32 cm^2
4. 20 cm^2

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

1. 20 cm^2
2. 32 cm^2
3. 40 cm^2
4. 80 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.

1. 2/3
2. 4/3
3. 10/3
4. 16/3

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?

1. 15 - 12.5 sin(1.2)
2. 25 pi - 15 - 12.5 sin(1.2)
3. 25 pi - 15 + 12.5 sin(1.2)
4. 25 pi + 15 - 12.5 sin(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.

1. 7 sin(1) cm
2. 14 sin(2) cm
3. 7 sin(2) cm
4. 14 sin(1) cm