Circles, Arcs, and Sectors · Sub-skill drill
Arc Length and Sector Area
An arc of a circle subtends a central angle. If the angle is in degrees, arc length is (θ/360) × 2πr and sector area is (θ/360) × πr². If the angle is in radians, arc length is rθ and sector area is (1/2)r²θ. SAT questions test both forms; the radian form is increasingly common on the digital SAT. The fraction θ/360 (or θ/2π) is the same fraction of the full perimeter or area that the sector represents.
How this sub-skill is tested on the SAT
An arc of a circle subtends a central angle. If the angle is in degrees, arc length is (θ/360) × 2πr and sector area is (θ/360) × πr². If the angle is in radians, arc length is rθ and sector area is (1/2)r²θ. SAT questions test both forms; the radian form is increasingly common on the digital SAT. The fraction θ/360 (or θ/2π) is the same fraction of the full perimeter or area that the sector represents.
This sub-skill sits inside the broader Circles, Arcs, and Sectors topic, which is part of the College Board's Additional Topics in Math content domain. Additional Topics in Math is the smallest official SAT Math domain by raw question count, but it carries outsized weight because the questions are concentrated at the harder end of each section. You will see roughly six of these per test, and they tend to separate students aiming for a 750 from students aiming for an 800. The domain covers right triangle trigonometry, circle theorems, volume formulas, complex number arithmetic, and the geometry of lines in the coordinate plane. Most of the formulas you need are listed at the start of the math section — but the test rewards students who have
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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A circle has radius 11. What is the area of a sector with central angle 45 degrees? Use π ≈ 3.14.
- A 380.13
- B 95.03
- C 47.52
- D 8.64
Worked solution
Answer: C — 47.52
Sector area = (angle / 360) × πr^2 = (45/360) × π(11)^2 = 47.52.
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A circle has radius 4. What is the area of a sector with central angle 135 degrees? Use π ≈ 3.14.
- A 18.85
- B 50.27
- C 9.42
- D 37.70
Worked solution
Answer: A — 18.85
Sector area = (angle / 360) × πr^2 = (135/360) × π(4)^2 = 18.85.
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A circle has radius 9. What is the area of a sector with central angle 60 degrees? Use π ≈ 3.14.
- A 84.82
- B 42.41
- C 9.42
- D 254.47
Worked solution
Answer: B — 42.41
Sector area = (angle / 360) × πr^2 = (60/360) × π(9)^2 = 42.41.
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A circle has radius 12. What is the area of a sector with central angle 180 degrees? Use π ≈ 3.14.
- A 226.19
- B 452.39
- C 37.70
- D 453.39
Worked solution
Answer: A — 226.19
Sector area = (angle / 360) × πr^2 = (180/360) × π(12)^2 = 226.19.
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A circle has radius 3. What is the area of a sector with central angle 90 degrees? Use π ≈ 3.14.
- A 7.07
- B 28.27
- C 4.71
- D 14.14
Worked solution
Answer: A — 7.07
Sector area = (angle / 360) × πr^2 = (90/360) × π(3)^2 = 7.07.
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A circle has radius 6. What is the area of a sector with central angle 30 degrees? Use π ≈ 3.14.
- A 18.85
- B 3.14
- C 9.42
- D 113.10
Worked solution
Answer: C — 9.42
Sector area = (angle / 360) × πr^2 = (30/360) × π(6)^2 = 9.42.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.