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.

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

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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.

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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.

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

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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.

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Joined together gives an inscribed regular decagon.

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There are 13 Archimedean Solids.

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