Circular Segment Area Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The area of a circular segment is calculated with A = (r²÷2) × (θ − sin θ), where θ is the central angle in radians. For a circle of radius 10 cm and an angle of 90°, the segment area is about 28.54 cm².
Explanation
A circular segment is the portion of a disk bounded by a chord and the arc it subtends — not to be confused with the circular sector, the "pie slice"-shaped portion bounded by two radii and an arc. The segment is obtained by subtracting from the circular sector the area of the triangle formed by the two radii and the chord: this is exactly what the "− sin θ" term does in the formula, since the area of that isosceles triangle is (r²÷2) × sin θ. When the central angle reaches 180°, the chord becomes a diameter and the segment coincides with an entire half-disk: the formula confirms this exactly, since sin(180°) = 0 removes the triangle term, leaving (r²÷2) × π, the area of a half-disk. This calculation arises in concrete situations such as the volume of liquid remaining in a partially filled horizontal cylindrical tank, or the surface of an arch-shaped window — cases where knowing only the arc length is not enough, since it is an area, not a length, that is sought.
Example: circle of radius 10 cm, central angle of 90°
Inputs
Radius: 10 cm. Central angle: 90° (π/2 radians).
Calculation
A = (10²÷2) × (π/2 − sin(90°)) = 50 × (1.5708 − 1) = 50 × 0.5708 ≈ 28.5398 cm².
Result
The area of this circular segment is about 28.54 cm².
Frequently asked questions
What is the difference between a circular segment and a circular sector?
The circular sector is bounded by two radii and an arc (the shape of a pie slice), while the circular segment is bounded by a single chord and the arc it subtends (the shape obtained by cutting a disk with a straight line). The segment is always obtained by subtracting a triangle from the corresponding sector.
Should the angle be entered in degrees or radians?
This calculator expects an angle in degrees, the most intuitive convention for most everyday uses — the conversion to radians, needed for the mathematical formula itself, is done automatically internally.
What does the segment area become when the angle approaches 360°?
It approaches the area of the whole disk (πr²): the chord tightens around a single point and the segment covers almost the entire circle, except for a thin, near-zero triangular slice. This is the symmetric opposite of an angle close to 0°, where the segment instead becomes a near-zero area.