Sector Area and Arc Length: Use the Angle Fraction

Updated September 26, 2026•Geometry & Mensuration

Find sector area and arc length by treating the central angle as a fraction of 360°, with examples and radius checks.

One idea, two formulas

A sector is a fraction of a full circle. If the central angle is θ degrees, the sector is θ/360 of the full circle.

Sector area = (θ/360) × πr² Arc length = (θ/360) × 2πr

Sector area example

Example: Radius 6 cm, angle 90°.

Area = (90/360) × π × 6² = 1/4 × 36π = 9π cm²

Arc length example

Same sector:

Arc = (90/360) × 2π × 6 = 3π cm

If the question gives a diameter, stop here first

The formulas use r, the radius. If the diameter is 14 cm, the radius is 7 cm. Forgetting to halve the diameter causes a factor-of-four error in area.

Exact vs decimal answers

If the question accepts an exact answer, leave π in the result. If a decimal is requested, substitute a calculator value for π at the end, not halfway through, to reduce rounding error.

Use the full circle as a reality check

A 90° sector is one quarter of a circle. Your area and arc length should therefore be one quarter of the full-circle values.

Try one yourself

For a 180° sector of radius 4, what is the numeric coefficient of π in the area?

Keep going

For another problem using related ideas, return to the Geometry & Mensuration page.