Circular Sector Area Calculator

Formula

A = (π × r² × θ) / 360

Area of a circular sector

Picture a pizza slice: that is a circular sector, the piece of a disk closed off by two radii and the arc running between them. With radius r and central angle θ, the area is A = (θ/360°)·π·r² if you measure θ in degrees, or A = (1/2)·r²·θ if θ is in radians. Put r = 5 m and θ = 60° into the first one and you get (60/360)·π·25 ≈ 13.09 m².

For the arc length, use L = r·θ with θ in radians. A quadrant (θ = 90°) takes up π·r²/4, and once θ reaches 360° you are back to the whole disk, π·r². One thing to keep straight: a sector is two radii plus an arc, while a segment is a chord plus an arc. To get the segment, take the sector and subtract the triangle that the two radii make.

Applications

Sectors are everywhere once you look. Pie charts set each slice's angle in proportion to its value, so 25% of a budget turns into a 90° sector. They also drive gear and sprocket design, the layout of fan and ventilator blades, radar sweeps, analogue speedometers, sundials and irrigation rigs with rotating sprinklers. Architects reach for them when designing spiral staircases and circular amphitheatres.

FAQ

Sector vs segment, what's the difference? A sector sits between two radii and an arc, like a pizza slice. A segment sits between a chord and an arc, which is the shape you are left with after slicing straight across the circle.

How to convert degrees to radians? Multiply the degree figure by π/180. That turns 60° into π/3 rad ≈ 1.047 rad.

Sector perimeter? Add the two radii to the arc and you have it: P = 2r + r·θ (θ in radians).

What does a 360° sector mean? It is just the whole disk, which brings you right back to A = π·r², the usual area-of-a-circle formula.