Area of a sector

Sector area and boundary

A sector occupies the same fraction of the disc as its angle occupies of a full turn. Multiply the disc area by that fraction to find the sector area. Below a full turn, its perimeter includes the arc and two radii. At a full turn, the region is the whole disc and the radii are not boundary edges. Area has square units; arc and perimeter have length units

A sector has radius 6 cm and an angle of two-thirds of a radian. Find its area in cm²

Original sector

\(\displaystyle r=6\,\mathrm{cm},\quad\theta=\frac23\,\mathrm{rad}\)

For that sector, find its perimeter in cm

A sector of radius 10 cm has area 35 cm². Find its angle in radians

Major sector and segment are different

Subtract the minor angle from a full turn to find the major-sector angle. A segment is instead bounded by a chord and an arc. The central triangle belongs to the sector but not to its minor segment; a sector with angle below a full turn uses two radii in its perimeter, while a segment perimeter uses one chord

A circle has radius 4 cm and minor angle π/3. Give the coefficient of π in the major sector area

Which boundary belongs to a circular segment?

Derive and check the formula

Derive the sector-area formula from a fraction of the whole disc. Check θ=π and θ=2π, then explain the units and why a diameter must first be halved. This derivation is ungraded

Model: a fraction of the disc

A=[θ/(2π)]πr²=½r²θ. A half-turn gives ½πr² and a full turn πr². The factor r² gives square units. If the given diameter is d, use r=d/2; substituting d for r quadruples the area