About the Sector Area Calculator
A sector is the “pizza slice” of a circle bounded by two radii and the arc between them; a segment is the part cut off by a straight chord across the same arc. This sector area calculator takes the radius and the central angle (in degrees or radians) and returns the sector area, the segment area, the arc length, the chord length and the sector perimeter.
Use it for geometry and trigonometry homework, for sizing a pie-shaped garden bed or patio, cutting a curved piece of fabric or sheet metal, working out how much of a round pizza or cake a slice is, or finding the area of a partial circular window.
Angles up to 360° (2π radians) are accepted. For angles larger than 180° the segment is the larger piece of the circle, and its area formula still holds because sin θ becomes negative.
With the default inputs, the sector area is 52.3599. Change any value above to recalculate instantly.
How to use the sector area calculator
- 1Enter the radius of the circle.
- 2Enter the central angle of the slice.
- 3Choose whether the angle is in degrees or radians.
- 4Read the sector area, segment area, arc and chord lengths.
Formula and method
A sector is a fraction of the whole circle equal to its angle divided by a full turn, so its area is (θ/360°) × πr², which in radians simplifies to ½r²θ. The arc length follows the same idea: rθ with θ in radians.
A segment is the sector minus the isosceles triangle formed by the two radii and the chord. That triangle has area ½r²·sin θ, so the segment area is ½r²(θ − sin θ). The chord length comes from splitting the triangle into two right triangles: c = 2r·sin(θ/2).
- r
- Radius
- θ
- Central angle (radians in the formulas)
- s
- Arc length
- c
- Chord length
Worked examples
Radius 10, 60° sector
A 60° slice is one sixth of the circle, so its area is 100π/6 ≈ 52.36. The two radii and the chord form an equilateral triangle, so the chord is exactly 10, and the segment outside that triangle is about 9.06.
Quarter circle in radians (r = 8, θ = π/2)
A quarter circle of radius 8 has area ½ × 64 × π/2 = 16π ≈ 50.27. Removing the right triangle of area 32 leaves a segment of about 18.27, and the chord is 8√2 ≈ 11.31.
Pizza slice: 14-inch pizza cut into 8 (r = 7, 45°)
Each 45° slice of a 14-inch pizza covers 49π/8 ≈ 19.24 square inches, one eighth of the pie, and has about 5.5 inches of crust along the arc.
Frequently asked questions
How do you find the area of a sector?+
Area = (θ ÷ 360) × πr² with θ in degrees, or ½r²θ with θ in radians. A 90° sector of radius 4 has area ¼ × π × 16 ≈ 12.57.
What is the difference between a sector and a segment?+
A sector is bounded by two radii and an arc, like a slice of pie. A segment is bounded by a chord and an arc, like the piece cut off by a straight line across the circle.
How do I find the area of a circular segment?+
Subtract the triangle from the sector: A = ½r²(θ − sin θ), with θ in radians. For r = 10 and θ = 60° this gives about 9.06 square units.
How do you convert degrees to radians?+
Multiply degrees by π/180. For example 60° = π/3 ≈ 1.0472 radians and 90° = π/2 ≈ 1.5708 radians.
How do I find the sector angle from the arc length?+
Divide the arc length by the radius to get the angle in radians (θ = s/r), then multiply by 180/π for degrees. You can then enter that angle here.