Case 1
- Radius
- 5
- Central angle
- 90°
7.134954
Open with these valuesCircular Segment Area Calculator
7.134954units²
Result: 7.134954 units²A segment is what a straight cut leaves behind: the sector minus the triangle back to the centre, ½r²(θ − sin θ) with θ in radians. Enter the radius in any length unit and the central angle in degrees; the area comes back squared in that unit.
Worked examples
Case 2
- Radius
- 10
- Central angle
- 60°
9.058607
Open with these valuesCase 3
- Radius
- 8
- Central angle
- 120°
39.307830
Open with these valuesHow it's calculated
A = ½ × r² × (θ − sin θ), θ in radians
- StepEnter the radius of the circle in any length unit.
- StepEnter the central angle the chord spans, in degrees.
- ResultRead the area in the square of that unit.
Reference table
| Radius, angle | Exact | Area |
|---|---|---|
| 3, 45 | 4.5(π/4 − √2/2) | 0.352311 |
| 5, 90 | 12.5(π/2 − 1) | 7.134954 |
| 10, 60 | 50(π/3 − √3/2) | 9.058607 |
| 5, 180 | 12.5π | 39.269908 |
| 8, 120 | 32(2π/3 − √3/2) | 39.307830 |
| 7, 270 | 24.5(3π/2 + 1) | 139.953530 |
Questions
How do you calculate the area of a circular segment?
Take the sector area and subtract the triangle from the two radii to the chord: ½r²(θ − sin θ). Radius 5 at 90° leaves 12.5(π/2 − 1), about 7.13 square units.
How is a segment different from a sector?
A sector is bounded by two radii and the arc, a segment by a chord and the arc. The segment is always the smaller of the two.
Why does the angle have to become radians?
The term θ on its own is an arc length in units of the radius, and only radians measure it that way. The calculator converts the degrees for you.
What happens at 180 degrees?
The chord passes through the centre, the triangle collapses to nothing and the segment is a half circle, ½πr².
Sources and last check
Information, not professional advice.