Area of Sector/Proof 2

Theorem

Let $\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$.

Let $BAC$ be the sector of $\CC$ whose angle between $AB$ and $AC$ is $\theta$.

Then the area $\AA$ of sector $BAC$ is given by:

$\AA = \dfrac {r^2 \theta} 2$

where:

$r = AB$ is the length of the radius of the circle

$\theta$ is measured in radians.

Proof

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From Area of Circle, the area of $\CC$ is $\pi r^2$.

From Measurement of Full Angle, the angle within $\CC$ is $2 \pi$.

The fraction of the area of $\CC$ within the sector $BAC$ is therefore $\pi r^2 \times \dfrac \theta {2 \pi}$.

Hence the result.

$\blacksquare$