Circular Sector Area

A sector is a pizza slice: the wedge cut from a circle by two radii. Its area is simply the slice's share of the whole pie — the fraction θ/(2π) of the full circle's πr² — which collapses to the tidy A = ½r²θ when the angle is in radians. Degree input is converted automatically, but the ½r²θ form itself only works in radians; plugging in degrees directly is the classic trap. The formula is meaningful for angles from 0 up to 2π, where the sector becomes the whole circle.

A concrete field application: an irrigation sprinkler set to sweep a 120° arc (θ = 2π/3 ≈ 2.094 rad) with a 10 m throw waters A = ½ × 10² × 2.094 ≈ 105 m² of lawn — just one-third of the 314 m² a full-circle sweep would cover, exactly as the slice picture predicts.