# Area Enclosed by First Turn of Archimedean Spiral

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## Theorem

Let $S$ be the Archimedean spiral defined by the equation:

- $r = a \theta$

The area $\AA$ enclosed by the first turn of $S$ and the polar axis is given by:

- $\AA = \dfrac {4 \pi^3 a^2} 3$

## Proof

\(\ds \AA\) | \(=\) | \(\ds \int_0^{2 \pi} \frac {\paren {a \theta}^2} 2 \rd \theta\) | Area between Radii and Curve in Polar Coordinatesâ€Ž | |||||||||||

\(\ds \) | \(=\) | \(\ds \intlimits {\frac {a^2 \theta^3} 6} 0 {2 \pi}\) | Primitive of Power | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\paren {2 \pi}^3 a^2} 6\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {4 \pi^3 a^2} 3\) | after simplification |

$\blacksquare$

## Historical Note

The Area Enclosed by First Turn of Archimedean Spiral was first determined by Archimedes in his book *On Spirals*.

His proof appears as Proposition $24$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)