Area of a Circluar Sector (in Radian)

Area of a Circluar Sector (in Radian)

How to find the area of a circular sector when the central angle is given in radian: formula, example, and its solution.

Formula

A = (1/2)*(r^2)*theta, A: area of a circular sector, r: radius of the sector, theta: measure of the sector's central angle in radian

A = (1/2)r2θ

A: area of a circular sector
r: radius of the sector
θ: radian measure of the arc's central angle

Area of a circular sector

Proof

Area of a Circluar Sector (in Radian): Proof of the Formula

Recall the area of a circular sector (in degree):

A = πr2⋅[(degree)/360]

Change π⋅[(degree)/360]
to (1/2)⋅[π/180]⋅(degree).

[π/180]⋅(degree) = θ (radian)

Radian measure

Example

Find the area of the sector. Radius: 6. The measure of the central angle: 3pi/2 (rad).

r = 6, θ = 2π/3

A = (1/2)⋅62⋅(2π/3)