Circle segment area calculator

What is a circle segment?

A circular segment is the region of a disc bounded by a chord and the arc the chord cuts off. Picture a full pie slice (a sector), then remove the triangular wedge that connects the two endpoints of the arc to the center — what is left is the segment. It is the curved “cap” sitting between the chord and the arc.

The segment depends on two numbers: the radius $r$ of the circle and the central angle $\theta$ subtended by the chord at the center. The angle can be given in degrees, radians, or gradians; this calculator converts internally.

Key concepts

• Radius (r) — the distance from the center of the circle to a point on its boundary.
• Central angle (θ) — the angle formed at the center by the two radii drawn to the endpoints of the chord.
• Chord — the straight line connecting the two endpoints of the arc.
• Arc — the curved boundary of the segment, opposite to the chord.
• Sector — the pie-slice region bounded by the arc and the two radii.
• Triangle — the isosceles triangle with two sides equal to $r$ and the included angle $\theta$.

How does the calculator work?

The segment is what is left when the triangle is removed from the sector:

With $\theta$ in radians, the sector area is $\frac{1}{2} r^2 \theta$ and the area of the isosceles triangle formed by the two radii is $\frac{1}{2} r^2 \sin\theta$. Subtracting one from the other gives the standard formula.

Formula

If $\theta$ is in radians:

If $\theta$ is given in degrees, it is first converted to radians with $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$ before being substituted into the formula.

Worked examples

Example 1: small segment, 60°

A circle has a radius of 10 cm. The chord cuts off a central angle of 60°.

Convert: $\theta_{\text{rad}} = 60° \cdot \frac{\pi}{180} = \frac{\pi}{3} \approx 1.0472$.

Example 2: semicircle, π radians

For a radius of 5 cm and a central angle of $\pi$ radians (180°), the chord is a diameter and the segment is exactly half the disc:

Example 3: quarter-circle minus triangle, 90°

For a radius of 10 cm and a central angle of 90°:

This matches the intuition: the quarter sector has area $25\pi \approx 78.54$ cm², the right triangle has area $50$ cm², and the difference is the segment.

Practical uses

• Engineering — computing cross-sectional areas of partially filled circular tanks or pipes for fluid flow problems (this is the same calculation used by the circle area calculator when only a portion is filled).
• Construction and architecture — sizing windows, arches, and recessed details where the curved cap of a circle is a design element.
• Manufacturing — quoting material for stamped, cut, or machined parts shaped like a circular cap.
• Civil engineering — estimating earthwork volumes for circular channel cross-sections that are not full.
• Geometry and trigonometry — verifying the relationship to the circle sector area calculator and the chord length calculator.

Notes

• The angle must be positive. A 0° angle gives a degenerate segment with zero area.
• For $\theta = 2\pi$ (360°), the formula returns the area of the full circle.
• The “minor” segment corresponds to angles below 180°. For angles above 180°, the formula gives the larger “major” segment that includes the center.
• The radius and the area units must be consistent: a radius in metres produces an area in square metres. The unit selector reconverts the result automatically.
• The result is exact up to the precision of $\pi$ and the sine function; rounding errors are negligible for everyday use.