Last updated: 2026-05-19

The Circle Sector

A circle sector is the "pie slice" region enclosed by two radii and the arc between them. Given a radius r and a central angle θ, this calculator computes the sector area, arc length, chord length, and sector perimeter — with automatic conversion between degrees, radians, and gradians.

Formulas

Let θ be the central angle in radians and r the radius:

Why these formulas work

Area — A full circle (θ = 2π) has area πr². A sector spans a fraction θ/(2π) of that full circle, so:

A = (θ / 2π) × πr² = ½ r² θ

Arc length — The circumference of a full circle is 2πr. The arc is the same fraction of the circumference:

l = (θ / 2π) × 2πr = r θ

Chord — Drop a perpendicular from the circle's centre to the chord; it bisects both the chord and the angle θ. Each half-chord is r sin(θ/2), so the full chord is c = 2r sin(θ/2).

Worked example

A circular fountain has a radius of 3.5 m. The decorative tile section covers a central angle of 72°. What area do the tiles cover, and how long is the curved edge?

Convert the angle: 72° = 72 × π/180 = 2π/5 ≈ 1.2566 rad.

• Area: A = ½ × 3.5² × 1.2566 = ½ × 12.25 × 1.2566 ≈ 7.696 m²
• Arc length: l = 3.5 × 1.2566 ≈ 4.398 m
• Chord: c = 2 × 3.5 × sin(36°) ≈ 7 × 0.5878 ≈ 4.115 m
• Perimeter: P = 4.398 + 7 = 11.398 m

At a tile cost of £45/m², the section costs roughly 7.696 × 45 ≈ £346.