Circular Segment Area CalculatorFree online math calculator — instant results, no sign-up
Calculate the area between a chord and its circular arc.
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About the Circular Segment Area Calculator
The Circular Segment Area Calculator finds the area enclosed by a chord and its circular arc using the circle’s radius and the arc’s central angle. It distinguishes this curved segment from a sector, which is bounded by two radii and an arc. With a radius of 5 and a central angle of 90 degrees, the segment area is 7.1349540849 square units.
- Radius
- The distance from the circle’s center to its circumference, entered in a chosen length unit such as centimeters or meters.
- Central angle (degrees)
- The angle swept out at the circle’s center by the arc bounding the desired segment, measured in degrees.
Formula and method
θ = α × π / 180; A = (r² / 2)(θ − sin θ). For r = 5 and α = 90: A = (25 / 2)(π / 2 − 1) ≈ 7.1349540849.
- AArea of the circular segment, in square units.
- rRadius of the circle, in the chosen length unit.
- αCentral angle entered in degrees.
- θCentral angle converted to radians; sin θ must be evaluated in radians.
- πThe circle constant pi, approximately 3.141592653589793.
Convert the entered angle from degrees to radians, then evaluate A = (r² / 2)(θ − sin θ). For angles up to 180 degrees, this subtracts the triangle formed by the two radii and the chord from the corresponding sector. The same formula gives the major segment for angles above 180 degrees because the sine term becomes negative. Using the defaults, θ = 90 × π / 180 = π / 2 and sin θ = 1, so A = 12.5 × (π / 2 − 1) ≈ 7.1349540849 square units.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Radius5
- Central angle (degrees)90
Results
- Result7.1349540849
How to use the Circular Segment Area Calculator
- 1Enter the circle’s radius in Radius; use 5 to reproduce the default example.
- 2Enter the angle swept out by the desired arc in Central angle (degrees); use 90 for the default example.
- 3Check that the radius is positive and the angle describes the intended minor or major segment within one full turn.
- 4Run the calculation to convert the angle to radians and apply the circular segment area formula.
- 5Read the Result as an area: the default inputs give 7.1349540849 square units.
Reading your result
The Result row reports the area between the chord and the arc specified by the central angle, not the area of the entire sector or circle. For Radius = 5 and Central angle (degrees) = 90, Result = 7.1349540849. Its units are the square of the radius unit: a radius entered in centimeters produces an area in square centimeters.
Limitations
The formula assumes a perfect circle, a straight chord, a positive radius, and an angle from 0 to 360 degrees describing a single arc. Angles of 0 and 360 degrees are limiting cases with areas of zero and the full circle, respectively. Angles outside this range do not describe a single circular segment under this convention. Displayed decimals are rounded, and very small angles can lose numerical precision when θ and sin θ are subtracted. Accuracy also depends on the precision of the entered measurements.
Common questions
Why is the segment area smaller than the sector area for the default angle?+
At 90 degrees, the sector includes both the segment and the triangle formed by the two radii and the chord. With radius 5, the sector area is 25π/4 and the triangle area is 12.5; subtracting gives the segment area of 7.1349540849 square units.
How do I select the minor segment rather than the major segment?+
Use the central angle swept out by the desired arc. An angle below 180 degrees selects the minor segment, while an angle above 180 degrees selects the major segment. At 180 degrees, the chord is a diameter and either segment is a semicircle.
Should I convert my angle to radians before entering it?+
No. Enter the angle in degrees; the calculation converts it to radians before evaluating θ − sin θ. For the default 90-degree input, the converted angle is π/2 radians.
Can I calculate the segment area from the chord length alone?+
No. A chord length alone does not determine the circle or the segment area. If you also know the radius r, the minor central angle in radians is θ = 2 arcsin(c / (2r)), where c is the chord length and 0 ≤ c ≤ 2r. Convert that angle to degrees for this calculator; use its complement to 360 degrees if you want the major segment.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.