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Circular Segment Area (Radius and Height)

Calculate the area of a circular segment from the circle radius and the height (sagitta). It also finds the central angle, chord length, and arc length, with a diagram.

Input

Enter the circle radius r and the height (sagitta) h to compute the area of a circular segment, the region bounded by an arc and its chord. The central angle, chord length, and arc length are also shown.

Result

r = 10h = 4

Segment area

44.729522

Central angle

106.260205°

Chord length

16

Arc length

18.545904

Lengths use the same unit as the input, and the area is in that unit squared. The central angle is shown in degrees.

How it works

  • A circular segment is the region bounded by a chord and the arc it cuts off. The central angle is theta = 2 arccos(1 minus h/r).
  • The segment area is one half r squared times (theta minus sin theta), where theta is in radians.
  • The chord length is 2 times the square root of h times (2r minus h), and the arc length is r times theta.
  • The height h must be greater than 0 and at most the diameter 2r. When h equals 2r the segment becomes a half circle.
  • Lengths use the same unit as the input, and the area is in that unit squared.

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