Circular Segment Area Calculator
A circular segment is the region of a circle cut off by a chord (a straight line connecting two points on the circle). The segment consists of the chord on one side and the arc on the other side. This calculator computes the area of a circular segment from the radius and the central angle. It also calculates the chord length (the straight-line distance between the two endpoints of the arc) and the segment height, also called the sagitta, which is the perpendicular distance from the chord to the highest point of the arc. All calculations use standard trigonometric formulas for circular geometry.
Circular segment formulas
Area = (r^2 / 2) * (theta - sin(theta)) [theta in radians]
Chord length = 2 * r * sin(theta / 2)
Segment height = r * (1 - cos(theta / 2))
Example calculation
For a circular segment with radius 10 and central angle of 60 degrees (1.047 radians):
- Area = (10^2 / 2) * (1.047 - sin(1.047)) = 50 * (1.047 - 0.866) = 9.05 square units
- Chord length = 2 * 10 * sin(0.524) = 20 * 0.5 = 10.00 units
- Segment height = 10 * (1 - cos(0.524)) = 10 * 0.134 = 1.34 units
Circular segment calculator: frequently asked questions
What is a circular segment?
A circular segment is the region between a chord and the arc it subtends in a circle. It is the area cut off by a straight line (chord) from the circle. Unlike a sector, it does not include the two radii.
What is the formula for the area of a circular segment?
The area of a circular segment is: Area = (r^2/2) * (theta - sin(theta)), where r is the radius and theta is the central angle in radians. The angle in degrees must be converted to radians first.
What is the chord length in a circular segment?
The chord is the straight line that forms one boundary of the segment. The chord length is calculated as: Chord = 2 * r * sin(theta/2), where theta is the central angle in radians.
What is the segment height (sagitta)?
The segment height, also called the sagitta, is the perpendicular distance from the chord to the arc. It is calculated as: Height = r * (1 - cos(theta/2)), where theta is the central angle in radians.
What is the difference between a segment and a sector?
A sector includes two radii and the arc between them, while a segment only includes the chord and the arc. A sector has a triangular appearance, while a segment has a curved cap appearance.
Official sources
- Wolfram MathWorld: Circular Segment.
- ISO 80000-2: Mathematical signs and symbols.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.