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Calculate the area of a circular segment using chord length and height, or radius and central angle. Ideal for math education and engineering design.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
A circular segment is the region cut off from a circle by a chord and the arc connecting the chord's endpoints. This calculator takes a radius and a central angle in degrees, then returns the segment area to four decimal places. It does not take chord length and segment height as inputs, and it has no length-unit selector.
The calculation treats the segment as a circular sector minus the isosceles triangle formed by the two radii and the chord. If α is the entered angle in degrees, first convert it to radians, θ = α × π/180. The area is A = ½r²(θ − sin θ). Both θ and the sine function use radians in this expression. Angles below 180° describe a minor segment; 180° gives a semicircle; angles above 180° describe the larger segment corresponding to that arc.
θ = α × π/180
A = ½r²(θ − sin θ)
r = radius; α = entered central angle in degrees; θ = central angle in radians.
OpenStax explains the radian-based sector formula and the degree-to-radian conversion. Subtracting the triangle area from the sector gives the segment relationship used here.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Use a positive radius and keep the unit consistent with your measurement, such as inches, centimeters, or feet. There is no unit dropdown, so the calculator cannot infer or convert units.
Type a degree value such as 90, not a radian value such as π/2. The calculation converts the degree input internally before applying the trigonometric formula.
The result is displayed with four decimal places. Attach the square of the radius unit to the number: a radius entered in inches gives an area in square inches.
For example, with r = 10 and α = 90°, θ = π/2 and A = 50(π/2 − 1) ≈ 28.5398. If the radius was 10 feet, read the result as about 28.5398 square feet.
Use cases
See how the tool fits into real work and everyday tasks.
Compare a hand calculation with the displayed result while learning how a sector and its triangle form a segment. Check carefully that the angle reaches the sine function in radians.
When a drawing specifies the radius and angle of a circular cutout, estimate the planar area between the arc and chord, then record the squared measurement unit with the result.
Keep the radius fixed and calculate several degree angles to compare the area of different segments. Save the input pair with each result so the values remain interpretable.
Q&A
Find concise answers to common questions and confusing cases.
Enter 90. The input is in degrees. Entering approximately 1.57 would be interpreted as 1.57 degrees, not π/2 radians.
Not directly in this calculator. It requires a radius and central angle in degrees. If you know chord length and height, determine the matching radius and angle separately, then enter those values.
It selects the larger segment associated with the corresponding central arc. At 180° the segment is a semicircle. Use the angle that matches the arc bounding the region you want.
The tool returns a numeric area and does not ask for a unit. Since the formula squares the radius, interpret the result using the square of whatever length unit you entered.
Notes
Review scope, result limitations, and important precautions before use.
The input fields do not convert between inches, feet, meters, or other units. A mixed-unit radius or a result copied without its squared unit can lead to a misleading comparison. The formula assumes the boundary is an arc and chord from the same circle; it is not a model for an ellipse or an irregular measured outline.
The displayed value is rounded to four decimal places. Keep more precision in intermediate work if you will use the result in a later calculation, and independently check values used for engineering, fabrication, or other precision-critical decisions.
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