Circular Segment Calculator

Circular segment area and interpretation

The segment is the region between a chord and its chosen arc. Angles above 180° select a major segment. Height runs from the chord to the arc midpoint; arc length follows the curved boundary.

To find the radius, circumference, and full disk area of a circle, use the Circle Calculator and follow its own input conventions.

The formula

Convert the angle to radians and subtract the signed triangle contribution from the sector area. A small-angle series preserves shallow-segment area when direct subtraction would lose precision.

How to use this calculator

1. Enter a positive circle radius in one length unit.
2. Enter the selected arc’s central angle, strictly between 0° and 360°.
3. Read area in squared units and compare chord, height, and curved arc length.

Circular Segment input conventions

A circular segment is bounded by one chord and one arc. A sector instead extends to the circle’s center along two radii, so its area is larger for the same minor angle. The angle selector accepts values below or above 180 degrees. A value above 180 selects the major region and its longer arc, while the chord remains the straight line joining the arc endpoints.

Interpreting circular segment area

For a semicircle, the chord is a diameter, the height equals the radius, and the area is half of the full disk. Segment height is measured from the chord toward the chosen arc midpoint. The formula internally converts degrees to radians. At extremely small angles, a series expansion prevents loss of the small positive area when subtracting nearly equal values; measurement uncertainty can still dominate a shallow segment.

Assumptions & limitations

What this calculation assumes

• The angle selects the arc bounding the desired segment, in a flat circle of fixed radius.

What to keep in mind

• Radius and central-angle inputs only; not a partially filled tilted cylinder model.

Common questions