Area of Annulus as Area of Rectangle

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Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$.

The area of $A$ is given by:

$\map \Area A = 2 \pi \paren {r + \dfrac w 2} \times w$

where $w$ denotes the width of $A$.

That is, it is the area of the rectangle contained by:

the width of $A$
the circle midway in radius between the inner radius and outer radius of $A$.


\(\ds \map \Area A\) \(=\) \(\ds \pi \paren {R^2 - r^2}\) Area of Annulus
\(\ds \) \(=\) \(\ds \pi \paren {\paren {r + w}^2 - r^2}\) Definition of Width of Annulus
\(\ds \) \(=\) \(\ds \pi \paren {r^2 + 2 r w + w^2 - r^2}\) Square of Sum
\(\ds \) \(=\) \(\ds \pi \paren {2 r + w} w\) simplifying
\(\ds \) \(=\) \(\ds 2 \pi \paren {r + \frac w 2} \times w\) extracting $2$

The result follows from Area of Rectangle.