= Circular sector
{title2=$A=\theta R^2/2$}
{wiki}
A <circular sector> is the portion of a disc between two radii and the connecting <circle> arc. If its <radius> is $R$ and its central <angle> is $\theta$ <radians>, its <area> is
$$
\boxed{A=\frac12\theta R^2.}
$$
Indeed, <polar coordinates> give $A=\int_0^\theta\int_0^Rr\,dr\,d\phi$. The full disc corresponds to $\theta=2\pi$, a <semicircle> to $\theta=\pi$, and a quarter-disc to $\theta=\pi/2$. In a solid with horizontal sector cross-sections, <radius> $R(z)$ and <angle> $\theta(z)$, its <volume> is $\tfrac12\int\theta(z)R(z)^2\,dz$. Holding the <angles> and heights fixed while scaling every <radius> by a factor $s$ scales that <volume> by $s^2$.
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