Arc length and sector area

A sector is a pizza slice of a circle — scale the full circumference and area by θ/360°.

Key formulas

If the central angle is 90°, you take one quarter of the full circle formulas. The same idea works for any θ.

r=7, θ=60°. Arc = 2π·7·(60/360)= (7π)/3. Sector area = π·49·(1/6)= (49π)/6.

In radian measure the formulas become rθ and ½r²θ. Do not mix degree formulas with radian angles.

Problem. A sector has radius 7 cm and central angle 60°. Find the arc length and sector area in terms of π.

Solution. Arc = 2πr × (60/360) = 14π × (1/6) = 7π/3 cm. Area = πr² × (60/360) = 49π × (1/6) = 49π/6 cm².

θ=360° recovers the full circumference and area.

Sector r=10, θ=90°. Arc length is:

A segment is the region between a chord and its arc; subtract the triangular part from the sector.