SAT Math · Additional Topics in Math
Circles, arcs, and sectors
Circle equations, arc length, sector area, and central angles.
What's tested in this subtopic
Items test the standard circle equation (x − h)² + (y − k)² = r², arc length and sector area as fractions of the full circle, inscribed and central angles, and tangent-chord relationships.
Tactics that actually move your score
Arc length and sector area are both fractions of the full circle: arc = (θ/360°) × 2πr, sector area = (θ/360°) × πr². If the angle is in radians, arc = rθ and sector area = ½r²θ. Inscribed angle is half the central angle that subtends the same arc. The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy A circle has radius 10. What is its area, in terms of π?
- Easy A circle has radius 8. What is its area, in terms of π?
- Medium A circle has radius 8. What is its area, in terms of π?
- Medium A circle has radius 7. What is its area, in terms of π?
- Medium A circle has radius 9. What is its area, in terms of π?
- Hard A circle has radius 6. What is its area, in terms of π?
- Easy A circle has radius 12. What is its area, in terms of π?
- Easy A circle has radius 8. What is its area, in terms of π?
- Medium A circle has radius 5. What is its area, in terms of π?
- Medium A circle has radius 5. What is its area, in terms of π?
- Medium A circle has radius 7. What is its area, in terms of π?
- Hard A circle has radius 3. What is its area, in terms of π?
- Easy A circle has radius 12. What is its area, in terms of π?
- Easy A circle has radius 9. What is its area, in terms of π?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in SAT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.