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Pure · Trigonometry

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Arc length and sector area

The radian formulas for arc length, sector area and segment area, where they come from, and using them in reverse — finding a radius or angle from a given perimeter or area, and the perimeters of sectors and segments.

Pure · Trigonometry

Arc length and sector area

The radian formulas for arc length, sector area and segment area, where they come from, and using them in reverse — finding a radius or angle from a given perimeter or area, and the perimeters of sectors and segments.

Why it works

Radians make circle measurements clean, because a radian is defined so that the arc length equals the radius times the angle. Everything here is just "what fraction of the whole circle is this sector?"

Arc length. A full turn is 2π2\pi radians and sweeps the whole circumference 2πr2\pi r. An angle θ\theta is the fraction θ2π\dfrac{\theta}{2\pi} of a full turn, so the arc is that fraction of the circumference: s=θ2π×2πr=rθ.s = \frac{\theta}{2\pi}\times 2\pi r = r\theta.

Sector area. The same fraction of the whole area πr2\pi r^2: A=θ2π×πr2=12r2θ.A = \frac{\theta}{2\pi}\times \pi r^2 = \tfrac12 r^2\theta.

Segment area. A segment is the region between a chord and its arc — it's the sector with the triangle cut off. The triangle is formed by the two radii with included angle θ\theta, so its area is 12r2sin⁡θ\tfrac12 r^2\sin\theta. Therefore Asegment=12r2θ−12r2sin⁡θ=12r2(θ−sin⁡θ).A_{\text{segment}} = \tfrac12 r^2\theta - \tfrac12 r^2\sin\theta = \tfrac12 r^2(\theta - \sin\theta).

Perimeters. The perimeter of a sector is the two straight radii plus the curved arc: P=2r+rθ=r(2+θ)P = 2r + r\theta = r(2+\theta). The perimeter of a segment is the arc plus the chord, and the chord (from the isosceles triangle) is 2rsin⁡θ22r\sin\dfrac{\theta}{2}, so P=rθ+2rsin⁡θ2P = r\theta + 2r\sin\dfrac{\theta}{2}.

Every one of these needs θ\theta in radians. If a question gives the angle in degrees, convert it first (×π180\times\frac{\pi}{180}). And the formulas run backwards just as well: given an area or a perimeter you can solve for rr or θ\theta — often landing on a quadratic in rr with two valid answers.