SAT Math: Circle Arcs, Sectors, and Radians Practice Questions
Connect central angles with arc lengths and sector areas, and convert between degrees and radians. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
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1. A circle has radius 10. What is the length of the minor arc subtended by a 90-degree central angle?
Explanation. A 90-degree angle is one fourth of a full turn. Its arc is one fourth of the circumference 2pi(10), so the length is 5pi.
2. A sector has radius 12 and central angle 60 degrees. What is its area?
Explanation. The sector is 60/360 = 1/6 of the circle. Its area is (1/6)pi(12)^2 = 24pi square units.
3. An arc of length 9 belongs to a circle of radius 6. What is the corresponding central angle in radians?
Explanation. For an angle measured in radians, arc length s equals r times theta. Thus theta = s/r = 9/6 = 3/2 radians.
4. An angle of 3pi/4 radians equals how many degrees?
Explanation. Multiply radians by 180/pi: (3pi/4)(180/pi) = 135 degrees. The pi factors cancel.
5. A 270-degree arc in a circle has length 18pi. What is the circle's circumference?
Explanation. The arc is 270/360 = 3/4 of the circumference C. Solve (3/4)C = 18pi to obtain C = 24pi.
6. A sector has area 12pi in a circle of radius 6. What is its central angle in degrees?
Explanation. The entire circle has area 36pi. The sector occupies 12pi/36pi = 1/3 of it, so its central angle is (1/3)(360) = 120 degrees.
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FAQ
When can I use s = r times theta?
Use this formula when theta is in radians. Convert degrees to radians first, or use the fraction of the full circumference.
Do sectors and their arcs use the same fraction?
Yes. A central angle that is one third of a full turn produces one third of the circle's circumference and one third of its area.