Quick answer
A circular segment has area (r² ÷ 2)(θ − sin θ), with θ in radians. For r = 2 and θ = 180° the segment is a half circle with area 6.2832.
Enter the radius and central angle in degrees. The calculator returns the segment area.
Related tools: Area of a Circle Calculator: Area of a circle from its radius, diameter or circumference. Area Calculator: Area of common two-dimensional shapes. Circle Calculator: Radius, diameter, circumference and area of a circle from any one of them.
How to use the Circle Segment Area Calculator
- Enter radius (above 0).
- Enter central angle (°) (between 0 and 360).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Radius | A number | > 0 |
| Central angle | A number, in ° | > 0, ≤ 360 |
Formula and method
Formula
A = (r² ÷ 2)(θ − sin θ), θ in radians
IEEE-754 double precision; shown to at most 6 decimal places.
Sector area minus the triangle.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Radius = 2; Central angle = 180.
| Result | Value |
|---|---|
| Segment area | 6.283185 |
Example 2
Inputs: Radius = 10; Central angle = 90.
| Result | Value |
|---|---|
| Segment area | 28.539816 |
The first line is the headline result (segment area). IEEE-754 double precision; shown to at most 6 decimal places.
Sector minus triangle
The segment is the sector (½r²θ) minus the triangle formed by the two radii and the chord (½r² sin θ). At 180° the triangle has zero area, so the segment equals the sector: ½π × 4 = 6.283.
Another example
For r = 2 and θ = 90° the segment is (4 ÷ 2)(1.5708 − 1) = 1.1416. Angle up to 360°. It's used for partly filled pipes and tanks, and for lens shapes.
How the result changes with radius
Holding every other input at the first worked example, this table changes only radius. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Radius | Segment area |
|---|---|
| 1 | 1.570796 |
| 2 | 6.283185 |
| 4 | 25.132741 |
| 8 | 100.530965 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Central angle: entering "0" shows "Central angle must be greater than 0." beside that field and no result.
Choosing between the Circle Segment Area Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Area of a Circle Calculator](/math/area-of-a-circle/) | i know the and value | Area |
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
| [Circle Calculator](/math/circle-calculator/) | i know the and value | Area |
| [Circle Diameter Calculator](/math/circle-diameter/) | i know the and value | Diameter |
| [Area of a Trapezoid Calculator](/math/area-of-a-trapezoid/) | base a, base b and height | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Angle up to 360°
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for central angle to confirm this.
Frequently asked questions
What is a circular segment?
The region between a chord and its arc.
What if the angle is 360°?
A 360° sector is the whole circle, so the area is πr².
Do I need radians?
Enter degrees; the tool converts.
What does the Circle Segment Area Calculator calculate?
Area of a circular segment from radius and central angle. It returns Segment area.
What formula does the Circle Segment Area Calculator use?
A = (r² ÷ 2)(θ − sin θ), θ in radians. Sector area minus the triangle.
What do I need to enter in the circle Segment Area Calculator?
You enter Radius and Central angle (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Angle up to 360°.
What is an example of a circle Segment Area result?
With radius 2 and central angle 180, the segment area is 6.283185.
Related calculators
- Area of a Circle Calculator — Area of a circle from its radius, diameter or circumference.
- Area Calculator — Area of common two-dimensional shapes.
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
- Circle Diameter Calculator — Diameter of a circle from radius, circumference or area.
- Area of a Trapezoid Calculator — Area of a trapezoid from its parallel sides and height.
- Surface Area Calculator — Surface area of a sphere, cube, cylinder, cone or box.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.