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Circular Sector Area

A=12r2θA = \frac{1}{2} r^{2} \theta

Worked example: r = 6 m, 60 deg slice → 6 pi = 18.84956 m^2 — press Try an example to run it live, then adjust anything.

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π=3.141592653589793\pi = 3.141592653589793Pi · exact
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Circular Sector Area explained

θrA

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.

Circular Sector Area formula

A=12r2θA = \frac{1}{2} r^{2} \theta
Where
  • AA= Sector area (m²)
  • rr= Radius (m)
  • θ\theta= Central angle (°)

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