Circles • Topic 5 of 7

Sector Area

A sector is a “pie slice” of a circle bounded by two radii and an arc. Its area is the same fraction of the whole circle’s area as its central angle is of 360°: sector area = (θ/360) × πr². A 90° sector is one-fourth of the circle’s area; a 180° sector is half. For radius 6 and a 120° sector, the area is (120/360) × π(6)² = ⅓ × 36π = 12π. As always, leave the answer in terms of π. Use πr² here (area), not 2πr — that distinction between sector area and arc length is exactly what the SAT checks.

A circle with a shaded pie-slice sector spanning about 120 degreesSector areaθSector area = (θ/360) × πr²

✅ Solved examples

1. Radius 6, central angle 120°. Find the sector area in terms of π.
(120/360)(π·36) = ⅓ × 36π = 12π.
2. Radius 4, central angle 90°. Find the sector area.
¼ × 16π = 4π.
3. Radius 10, central angle 180°. Find the sector area.
½ × 100π = 50π.
4. Radius 6, central angle 60°. Find the sector area.
⅙ × 36π = 6π.

✏️ Practice — try these, take hints as needed

1. Radius 8, central angle 90°. Find the sector area in terms of π.
(θ/360)πr².
¼ × 64π.
—
16π.
2. Radius 6, central angle 180°. Find the sector area.
½ × 36π.
—
—
18π.
3. Radius 12, central angle 120°. Find the sector area.
⅓ × 144π.
—
—
48π.
4. Radius 10, central angle 90°. Find the sector area.
¼ × 100π.
—
—
25π.
5. Radius 3, central angle 120°. Find the sector area.
⅓ × 9π.
—
—
3π.

📝 Topic test — 8 questions

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