Last updated: 2026-05-20

The circular segment

A circular segment is the region of a circle enclosed between a chord and the arc it subtends. Its size is fixed by two quantities: the radius of the circle and the central angle the chord subtends at the centre. From those two inputs the chord length, sagitta (height), arc length, and segment area all follow.

Segments and sectors

A circle can be divided in several ways. Two of the most common regions are the sector and the segment:

• Sector — the "pie slice" bounded by two radii and an arc, a wedge cut from the centre.
• Segment — the region between a single chord and the arc above it, a slice cut straight across without passing through the centre.

Every chord creates two segments: the minor segment (the smaller region, when the central angle θ < π) and the major segment (the larger region). When θ = π the chord passes through the centre and both segments are equal semicircles.

The sagitta

The sagitta — Latin for "arrow" — is the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It is the height of the segment.

For radius r and central angle θ (in radians):

$h = r\bigl(1 - \cos(\theta/2)\bigr)$

At θ = π (a semicircle) the sagitta equals the radius: the chord is a diameter and the arc reaches the point diametrically opposite the chord midpoint. At very small θ the sagitta approaches zero; at θ = 2π it approaches 2r (the full diameter).

Formulas

All formulas use θ in radians. The calculator handles degree and gradian input automatically.

Why the segment area formula works

The segment is the sector minus the isosceles triangle formed by the two radii and the chord.

• Sector area = ½r²θ
• Triangle area = ½r²sin θ (base = chord = 2r sin(θ/2), height = r cos(θ/2), product = r² sin(θ/2)cos(θ/2) = ½r²sin θ)
• Segment area = sector − triangle = ½r²θ − ½r²sin θ = ½r²(θ − sin θ)

Worked example

A circular window has radius r = 10 m and a horizontal chord that subtends a central angle θ = 90°.

Convert to radians: θ = π/2 ≈ 1.5708 rad.

• Chord = 2 × 10 × sin(45°) = 20 × 0.7071 ≈ 14.14 m
• Sagitta = 10 × (1 − cos(45°)) = 10 × 0.2929 ≈ 2.93 m
• Arc = 10 × π/2 ≈ 15.71 m
• Segment area = ½ × 100 × (π/2 − 1) ≈ 50 × 0.5708 ≈ 28.54 m²

For comparison, the full circle area is π × 100 ≈ 314.16 m². This 90° segment is about 9.1% of the full circle.