Circular Segment Area Calculator
A circular segment is the region of a disk cut off by a straight chord, the slice between the chord and the arc it subtends. Its area is found from the radius of the circle and the central angle that the chord spans, measured at the center. This calculator takes those two numbers and returns the segment area using the standard formula: half the radius squared, multiplied by the central angle in radians minus the sine of that angle. The angle must be in radians, where a full circle is two pi, a half circle is pi, and a right angle is pi over two. The formula works because the segment area equals the area of the circular sector minus the area of the triangle formed by the two radii and the chord. Engineers, draftsmen, tank and pipe designers and students use it to find the cross-sectional area of partially filled cylinders, arched openings and lens shapes. Enter your radius and angle to get the area immediately. Both inputs are left editable so you can model any chord position. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
A circular segment's area is 0.5 r squared (theta - sin theta) with theta in radians. For r = 10 and theta = 1.5 radians the area is 25.13.
Circular Segment Area formula
A = (1/2) r^2 (theta - sin theta)
A = segment area
r = circle radius
theta = central angle in radians
sin theta uses theta in radians
The segment is the sector minus the triangle between the two radii and the chord. The sector area is half r squared times theta, and the triangle area is half r squared times sine theta, so the difference gives the segment.
Worked example
Find the area of a circular segment with radius 10 and a central angle of 1.5 radians.
- Sector area = 0.5 x 10^2 x 1.5 = 75
- Triangle area = 0.5 x 10^2 x sin(1.5) = 50 x 0.997495 = 49.8747
- Segment area = 75 - 49.8747 = 25.13
The segment area is 25.13. These are the calculator's default inputs, so the result above matches the widget exactly.
Segment area for radius 10 at common angles
A = 0.5 x 100 x (theta - sin theta).
| Angle (rad) | Segment area |
|---|---|
| 0.50 | 2.06 |
| 1.00 | 7.93 |
| 1.50 | 25.13 |
| 2.00 | 54.53 |
| 3.14 | 156.92 |
Mathematical functions reference: US National Institute of Standards and Technology (NIST).
Circular Segment Area Calculator: frequently asked questions
What is a circular segment?
A circular segment is the area between a chord and the arc it cuts off. Picture slicing a circle with a straight line; the smaller piece bounded by that line and the curve is a segment. It differs from a sector, which is the pie-slice region bounded by two radii and an arc.
Why must the angle be in radians?
The formula mixes the angle directly with its sine, and that only works when the angle is in radians, the natural measure where arc length equals radius times angle. To convert degrees to radians, multiply by pi and divide by 180. A right angle of 90 degrees is pi over two, about 1.5708 radians.
How is the segment related to the sector?
The segment equals the sector minus the triangle formed by the two radii and the chord. The sector area is half the radius squared times the angle, and the triangle area is half the radius squared times the sine of the angle. Subtracting the triangle from the sector leaves the segment.
Can I use this for a partially filled tank?
Yes. The cross-section of liquid in a horizontal cylindrical tank is a circular segment. Find the central angle from the fill depth, then apply this formula to get the segment area, and multiply by the tank length for volume. The angle relationship depends on the depth and radius.
What is the segment area formula?
The area equals one half the radius squared times the quantity angle minus sine of the angle, with the angle in radians: A = 0.5 r squared (theta - sin theta). For radius 10 and angle 1.5 radians the area is about 25.13.
Official sources
- Mathematical functions and computational reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.