Segment Area Calculator

Enter a circle's radius and the central angle of the chord to get the circular segment's area, chord length, arc length, and sagitta (height).

Quick Facts

Segment area formula
A = (r²/2)(θ − sinθ)
θ must be in radians; the calculator converts degrees for you.
Chord length
c = 2r × sin(θ/2)
The straight-line distance across the segment's base.
Sagitta (height)
h = r(1 − cos(θ/2))
Distance from the chord's midpoint to the arc.

Your Results

Calculated
Segment Area
-
A = (r²/2)(θ − sinθ)
Chord Length
-
c = 2r × sin(θ/2)
Arc Length
-
L = r × θ
Sagitta (Height)
-
h = r(1 − cos(θ/2))

Ready

Enter a radius and central angle, then press Calculate.

Formula and Method for Segment Area

A circular segment is the region of a circle cut off by a chord — the area between the chord and the arc it subtends. Given the circle's radius r and the central angle θ (in radians) subtended by the chord, the segment's area is A = (r²/2)(θ − sinθ). This calculator also derives the chord length, arc length, and sagitta (height) of the segment from the same two inputs.

How the calculation works

Enter the radius and the central angle, choosing whether the angle is in degrees or radians. The calculator first converts the angle to radians if needed, then applies A = (r²/2)(θ − sinθ) to get the segment area. The chord length is c = 2r × sin(θ/2), the arc length is L = r × θ, and the sagitta — the height of the segment, measured from the chord's midpoint to the arc — is h = r(1 − cos(θ/2)). All of these follow from the geometry of the circular sector minus the triangle formed by the two radii and the chord.

Common mistakes

  • Degrees vs. radians: the formula A = (r²/2)(θ − sinθ) requires θ in radians. Using degrees directly without converting gives a wildly wrong area — this calculator handles the conversion for you when you choose "Degrees."
  • Segment vs. sector: a circular sector is the full pie-slice area, r²θ/2. The segment is smaller — it excludes the triangular wedge between the two radii and only counts the area beyond the chord.
  • Angle range: the central angle must be strictly between 0° and 360° (0 and 2π radians). At 360° there is no chord left to cut a segment from — the "segment" would be the whole circle.

Real-world applications

  • Engineering: finding the cross-sectional area of liquid in a partially filled horizontal cylindrical tank uses the segment area formula directly.
  • Architecture: segmental arches and window openings use segment geometry to size glazing or material.
  • Manufacturing: cutting a circular segment ("D-shape") from sheet stock relies on the same area and chord formulas.
  • Surveying and construction: sagitta measurements check the curvature of arcs, roads, and rail track.

Frequently Asked Questions

What is the formula for the area of a circular segment?
A circular segment is the region between a chord and the arc it cuts off. Its area is A = (r²/2)(θ − sinθ), where r is the circle's radius and θ is the central angle in radians. For a semicircle (θ = π), this reduces to half the circle's area, ½πr².
How do I find the central angle if I only know the chord length?
Given a chord length c and radius r, the central angle in radians is θ = 2 × arcsin(c / (2r)). Enter that value (converted to degrees if needed) as this calculator's angle input.
What is the sagitta of a circular segment?
The sagitta, or height, is the distance from the midpoint of the chord to the midpoint of the arc: h = r(1 − cos(θ/2)). It is also related to the chord and radius by h = r − √(r² − (c/2)²).

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