# Trisecting the Angle/Archimedean Spiral

## Theorem

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of an Archimedean spiral.

## Construction

In the figure, the blue line is an Archimedean spiral.

Let $\angle AOB$ be the angle to be trisected, where $OB$ is on the polar axis.

Trisect the segment $OA$ such that $OC = \dfrac 1 3 OA$.

Draw a circle with $O$ as the center and $OC$ as the radius.

Let $D$ be the intersection of the circle and the spiral.

Then $\angle DOB = \dfrac 1 3 \angle AOB$.

## Proof

Let the equation of the Archimedean spiral be $r = a \theta$.

Then:

 $\ds \angle DOB$ $=$ $\ds \frac {OD} a$ $\ds$ $=$ $\ds \frac {OA/3} a$ $\ds$ $=$ $\ds \frac 1 3 \angle AOB$

$\blacksquare$

## Historical Note

Use of the Archimedean spiral to trisect an angle was a standard technique for mathematicians following Archimedes.