Area of a Sector Calculator

Area of a sector formula

A sector is the wedge of a circle enclosed by two radii and the arc between them, like a slice of pie. Its area is the same fraction of the whole disc as its central angle is of a full turn.

A = (α ÷ 360) × π × r²α is the central angle in degrees, r is the radius.

When the angle is given in radians, the fraction of the circle is θ ÷ 2π, and the formula simplifies:

A = ½ × r² × θθ in radians. Radians = degrees × π ÷ 180.

Setting α = 360° recovers the full circle, π × r², and α = 180° gives a half-disc.

How to use the calculator

1. Enter the Radius and pick its unit (mm up to miles).
2. Enter the Angle (α) in degrees, from 0 to 360.
3. Press Calculate area. The result is in the radius unit, squared, rounded to six decimals.

The angle is the one at the centre of the circle, between the two radii. If you have a fraction of a circle instead, multiply it by 360 to get degrees: one eighth of a circle is 45°.

Worked examples

A quarter circle. With r = 5 cm and α = 90°, A = (90 ÷ 360) × π × 25 = 0.25 × 78.539816 = 19.634954 cm².

A pizza slice. A 16 inch pizza has a radius of 8 in. Cut into 8 equal slices, each slice has a 45° angle, so its area is (45 ÷ 360) × π × 64 = 25.132741 in², one eighth of the 201.06 in² total.

A fractional angle. For r = 3.5 m and α = 47.5°, A = (47.5 ÷ 360) × π × 12.25 = 5.077817 m².

Working backward. To find the angle that gives an area of 30 on a radius of 6, rearrange the formula: α = 360 × A ÷ (π × r²) = 360 × 30 ÷ 113.097336 ≈ 95.49°.

Sector area for common angles

Areas for a circle of radius 1. For any other radius, multiply by r².

Arc length, perimeter and segments

The curved edge of a sector is its arc, and its length is the same fraction of the circumference.

arc length = (α ÷ 360) × 2 × π × r

For r = 5 and α = 90°, the arc is 7.853982 long and the full outline of the sector (two radii plus the arc) is 10 + 7.853982 = 17.853982. A segment is the region between a chord and its arc, not the pie-slice shape. Its area is the sector minus the triangle formed by the two radii and the chord: ½ × r² × (θ − sin θ). With r = 5 and a 90° angle, that is 7.134954.

To find the area of the full circle, use the circle area calculator. For other shapes, the multi-shape area calculator lets you switch between them.

Common mistakes

• Typing radians into the degree box. An angle of π ÷ 2 entered as 1.5708 would be read as a tiny 1.5708°.
• Using the diameter as the radius. The area would be four times too large.
• Dividing by 180 instead of 360. That doubles the answer, because 180 is the half-circle angle.
• Confusing sector with segment. A sector touches the centre of the circle; a segment is cut off by a chord.