There are five regular polyhedrons, also called the Platonic solids. Each is comprised of the same regular polygon. Only three regular polygons make up all of these solids: equilateral triangle, square and regular pentagon. There is another class of polyhedra which are comprised of different regular polygons. These are the semi-regular polyhedra, otherwise known as the Archimedean Solids.

They are named after Archimedes, the Greek Philosopher.

Unlike the Platonic solids, the Archimedean solids include the following additional regular polygons:

Hexagon, Octagon & Decagon. We have investigated how to construct an inscribed regular hexagon in a circle using a compass and straightedge.

##### Regular Octagon

To inscribe a regular octagon in a circle we shall return to the inscribed square ABCD.

We need to bisect the arc AB. The arcs cross at X and Y.

The line XY meets the circle at Z. With the compass at Z, adjusted to the length AZ, we can now draw 8 arcs around the circle to establish the vertices of a regular octagon. Joined together gives an inscibed regular octagon.

##### Regular Decagon

A decagon is a 10 sided polygon. To inscribe a regular decagon in a circle we shall return to the inscribed pentagon ABCDE.

We need to bisect the arc AB. The arcs cross at X and Y. The line XY meets the circle at Z.

With the compass at Z, adjusted to the length AZ, we can now draw 10 arcs around the circle to establish the vertices of a regular decagon.

Joined together gives an inscribed regular decagon.

There are 13 Archimedean Solids.

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