# Area Enclosed by First Turn of Archimedean Spiral

## 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 Coordinates‎ $\ds$ $=$ $\ds \intlimits {\frac {a^2 \theta^3} 6} 0 {2 \pi}$ 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 $24$.