Sector and Arc Length Calculator

How it works

A sector is the pie-slice between two radii and the arc that joins them. Its size is set by the radius and the central angle: the share of the full circle it takes is the angle divided by 360° (or by 2π radians), so the arc is that share of the circumference and the area is that share of the circle's area.

In radians the formulas are simplest: arc length is r × θ and sector area is ½ × r² × θ. An angle in degrees is converted by multiplying by π ÷ 180. The sector's perimeter is the arc plus the two radii, and the chord, the straight line between the arc's ends, is 2r × sin(θ ÷ 2).

The presets set the angle to 180° for a semicircle and 90° for a quarter circle. At 360° the sector is the whole circle: the arc equals the circumference, the area equals π r², and the chord collapses to zero.

Formula

θ (radians) = degrees × π ÷ 180
arc length  = r × θ            = 2 × π × r × (degrees ÷ 360)
sector area = ½ × r² × θ       = π × r² × (degrees ÷ 360)
perimeter   = arc length + 2 × r
chord       = 2 × r × sin(θ ÷ 2)

Example

A 60° sector of a circle with a 6 ft radius is one sixth of the circle. Its arc is 2π × 6 ÷ 6 = 2π ≈ 6.28 ft, its area is π × 36 ÷ 6 = 6π ≈ 18.85 ft², its perimeter is 6.28 + 12 ≈ 18.28 ft, and its chord is 2 × 6 × sin 30° = 6 ft, the same as the radius, because a 60° sector's chord and two radii form an equilateral triangle.

A quarter circle of radius 2 ft has an arc of π ≈ 3.14 ft, an area of π ≈ 3.14 ft², and a chord of 2√2 ≈ 2.83 ft.

Assumptions and limitations

• The central angle is between 0 and 360° (0 and 2π radians); 360° is the full circle.
• The radius is in the unit you choose; the unit labels the result and does not convert anything. The area is in that unit squared.
• The chord is the straight distance between the two ends of the arc, and the perimeter is the arc plus the two straight edges, not the chord.
• Results are rounded for display only.

Frequently asked questions