Calculators
Circular Segment Calculator
Calculate circular segment area, arc length, chord length, height, and perimeter from a radius and central angle in degrees.
Measure the region between a chord and an arc.
A circular segment is bounded by a chord and its circular arc. Enter circle radius r and the central angle of the desired arc in degrees; t = θπ/180 converts to radians. Segment area = r²(t − sin(t))/2, arc length = rt, chord length = 2r sin(t/2), height (sagitta) = 2r sin²(t/4), and perimeter = arc + chord. For r = 5 and θ = 90°, area is approximately 7.13495408494 square units, chord length is 7.07106781187 units, and height is 1.46446609407 units. Angles below 180° give minor segments; angles above 180° give major segments. At 180° the segment is a semicircle; at 360° it is the full disk with zero chord, height 2r, and perimeter 2πr. Use one consistent length unit; area uses square units. Radius must be from 1e-50 to 1e50 and angle from 1e-10 to 360 degrees. Small angles use a series expansion to avoid cancellation in the area formula. Results use browser floating-point arithmetic rounded to 12 significant digits. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is the circular segment area formula?
For central angle t in radians and radius r, area = r²(t − sin(t))/2. Convert degrees to radians with t = θπ/180. The calculator uses a stable series for small angles.
How does a segment differ from a sector?
A segment lies between a chord and its arc. A sector lies between two radii and their arc. For a minor segment, subtract the triangle area from the sector area.
Can I calculate major segments?
Yes. Enter an arc angle above 180° through 360°. Height is measured from the chord to the midpoint of that chosen arc and exceeds the radius for major segments.
What happens at 180 or 360 degrees?
At 180° the result is a semicircle with height r and chord 2r. At 360° it is a full disk with height 2r, zero chord, and perimeter 2πr.
Which inputs and units are supported?
Use radius 1e-50 to 1e50 and angle 1e-10 to 360 degrees. Length results use the radius unit; area uses its square. Values stay in your browser. Copy and reset controls are included.