Circle Segment Calculator
Find the area, arc length, chord, height and angle of a circular segment from radius and angle, radius and height, or chord and height.
A circular segment is the region cut off from a circle by a chord: the area between the straight line and the arc. You need it for the liquid in a horizontal tank, the glass in an arched window, the cross-section of a culvert or the rise of a curved beam. Enter any of the supported pairs of measurements to get the area, arc length, chord, height and central angle.
How to use this tool
- Choose the two measurements you know: radius and angle, radius and height, chord and height, or radius and chord.
- Enter the values. The height (sagitta) is the distance from the middle of the chord to the arc.
- Read the segment area and the other dimensions; the other-segment area is the rest of the circle.
Formula
A = ½ r² (θ − sin θ) c = 2r sin(θ/2) h = r (1 − cos(θ/2)) s = rθ
θ is the central angle in radians. From a chord and height the radius is r = h/2 + c² / (8h). From a radius and height the angle is θ = 2 arccos((r − h) / r), which also covers segments larger than a semicircle (h > r). The area is the sector ½r²θ minus the triangle ½r² sin θ.
Example
An arch with a chord of 8 m and a rise of 2 m: r = 2/2 + 64/16 = 5 m. Then θ = 2 arccos(3/5) ≈ 1.8546 rad (106.26°), sin θ = 0.96, and A = ½ × 25 × (1.8546 − 0.96) ≈ 11.18 m². The arc is 5 × 1.8546 ≈ 9.27 m long.
Frequently asked questions
How do I find the volume of liquid in a horizontal tank?
Enter the tank radius and the liquid depth as the height to get the segment area, then multiply by the tank length. A depth of 0.5 m in a tank of radius 1 m gives about 0.614 m², so a 3 m long tank holds about 1.84 m³.
What is the sagitta?
It is the height of the segment, measured from the midpoint of the chord to the arc. Carpenters, bridge builders and lens makers use it to describe how deep a curve is.
What is the difference between a minor and a major segment?
A minor segment is smaller than half the circle (central angle under 180°); the major segment is the rest. The calculator gives both areas: the second one appears as the other-segment area.
Last updated: 2026-10-30