Area of Sector/Proof 2
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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:
Proof
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From Area of Circle, the area of $\CC$ is $\pi r^2$.
From Full Angle measures $2 \pi$ Radians, 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.
$\blacksquare$