Equilateral triangle inscribed in a circle

This page shows how to construct (draw) an equilateral triangle inscribed in a circle with a compass and straightedge or ruler. This is the largest equilateral triangle that will fit in the circle, with each vertex touching the circle. This is very similar to the construction of an inscribed hexagon, except we use every other vertex instead of all six.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Explanation of method

As can be seen in Definition of a Hexagon, each side of a regular hexagon is equal to the distance from the center to any vertex. This construction simply sets the compass width to that radius, and then steps that length off around the circle to create the six vertices of a hexagon.

But instead of drawing a hexagon, we use every other vertex to make a triangle instead. Since the hexagon construction effectively divided the circle into six equal arcs, by using every other point, we divide it into three equal arcs instead. The three chords of these arcs form the desired equilateral triangle.

Another way of thinking about it is that both the hexagon and equilateral triangle are regular polygons, one with double the number of sides of the other.

Proof

The image below is the final drawing from the above animation, but with extra lines and the vertices labelled.