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Circle Segment Calculator

Find the area of a circle segment.

Result

28.54

How it is calculated

The segment area is the sector area minus the triangle formed by the two radii and the chord: A = ½ × r2 × (θ − sin θ). r is the radius in any length unit and θ is the central angle, entered in degrees and converted to radians internally. The area comes back in the square of the unit you used, with two decimals. The radius must be above zero and the angle must be greater than 0 and at most 360; otherwise there is no result. Angles above 180 are accepted and give the larger segment; at 360 the segment is the whole circle, π r2. The tool returns only the area, not the chord length or arc length.

Worked example

Radius 10 and central angle 90 degrees. Convert the angle: 90 degrees = 1.5708 rad. sin 90 degrees = 1. Then A = 0.5 × 102 × (1.5708 − 1) = 50 × 0.5708 = 28.54 square units. The sector would be 78.54, and the triangle 50, so the segment is the 28.54 left between chord and arc.

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Common mistakes

  • Entering the angle in radians. The field is degrees, so typing 1.57 for a quarter circle is read as a very thin 1.57 degree sliver.
  • Confusing the central angle with the angle at the circumference, which is half as big for the same arc.
  • Entering the diameter instead of the radius, which makes the area four times too large.
  • Mixing up segment and sector. A sector (78.54 for this example) includes the triangle to the centre; the segment does not.
  • For a tank or sheet, using the depth of the liquid or the cut height as the angle. You must first find the central angle from the chord or depth.

About this calculator

This calculator finds the area of a circular segment — the smaller region cut off between a chord and the arc it subtends (different from a sector, which includes the two straight radii out to the center) — from the circle's radius and the central angle spanning that arc.

It is used by students distinguishing segment from sector problems in geometry, and in practical settings like calculating the cross-sectional area of liquid in a partially filled cylindrical tank lying on its side, or figuring out the material cut from a circular sheet when a straight edge is trimmed off — situations a simple sector formula cannot answer.

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Comments (1)

  • Editorial team

    If you know the chord's rise rather than the central angle, convert first: angle = 2 times arccos(1 minus h over r), where h is the height of the segment at its tallest point. A radius of 5 with a segment height of 1 gives about 73.74 degrees and an area of 4.09. That helps for arches, partly filled round tanks or cut-out panels. Which of these are you measuring?

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