How this calculator works
A sector is the slice of a circle cut by two radii, like a slice of pie. Its area is the same fraction of the whole circle that the central angle is of a full turn. With the angle in radians, that fraction is angle divided by 2pi, and the whole area is pi r squared, so the sector area simplifies to angle times r squared divided by two. The curved edge of the slice is the arc, and the two straight edges are radii. The calculator can recover the radius from a sector area or an arc length as well as from a radius or diameter, then reports every property of the slice. A zero angle describes a sector with no area, which is allowed.
Formula
Sector area A = θr²/2, where θ is in radians. Arc length L = θr. Chord c = 2r sin(θ/2). Sector perimeter P = L + 2r. Reversing the area formula gives r = sqrt(2A/θ), and reversing the arc formula gives r = L/θ.
Worked example
A 90 degree angle is π/2 radians. With a radius of 5 m, the sector area is (π/2) × 25 / 2 ≈ 19.635 m², the arc is about 7.854 m, the chord is about 7.071 m, and the perimeter is about 17.854 m. With a 60 degree angle and a diameter of 20 m, the radius is 10 m, the sector area is about 52.360 m² and the chord is exactly 10 m.
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Frequently asked questions
Why is the sector area not just angle times radius?
Angle times radius gives the arc length, which is a distance along the edge. Area grows with the square of the radius, so the sector area needs r squared and a factor of one half. For a full turn, the formula reduces to πr², the area of the whole circle.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, and includes the centre. A segment is the region between a chord and an arc, and does not reach the centre. The segment area is the sector area minus the triangle formed by the two radii and the chord.
Can the angle be larger than 180 degrees?
Yes, up to a full 360 degrees. A reflex sector is more than half the circle. Its area keeps growing with the angle, and at a full turn it equals the whole circle.
Which angle unit should I use?
Either degrees or radians is fine. The formula shown uses radians, and the calculator converts whatever unit you select before calculating. The angle result is displayed in the unit selector beside it.