Mathematics was one of many hobbies for the French General Napoleon. He was particularly interested in the study of geometry and its relationship with regular polygons.

In 1797, Napoleon explained the solutions to several geometry problems that he had learned while in Italy, to a couple of world famous mathematicians in Paris. Neither mathematician had heard of these solutions to the problems. One of the mathematicians reportedly remarked to Napoleon:

“General, we expect everything of you, except lessons in geometry!”

A regular polygon is a figure whose sides are all the same length and whose internal angles are all equal. The simplest regular polygon is the equilateral triangle, followed by the square. The early Greek Mathematicians named the remaining regular polygons after the names of their numbers.

5 6 7 8 9 10
Pente Hex Hepta Octo Ennea* Deca

 

*The Greek name for a regular polygon with nine sides was “enneagon”. It is now called a “nonagon”, derived from the Latin word “nonus” meaning “nine”.

Typically, regular polygons can be constructed using only a compass and straightedge by inscribing them inside a circle. That is, the circle circumscribes the polygon.

We will now investigate how to inscribe regular polygons inside a circle.

The equilateral triangle, square and regular pentagon are of particular interest as individually they each form the five regular polyhedrons, (often referred to as Platonic Solids).

Regular Hexagon

The sides of a regular hexagon are equal to the radius of the circle that circumscribes it. This then makes the construction of a regular hexagon extremely easy.

Draw a circle. Without adjusting the compass mark off 6 arcs around the circumference of the circle.

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Connect the six arc crossings results in an inscribed regular hexagon.

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Equilateral Triangle

In the previous post an equilateral triangle was constructed using a compass and straightedge. An equilateral triangle can be easily inscribed in a circle by connecting every second arc in the regular hexagon construction.

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Square

To inscribe a square we need to use four consecutive arc crossings used in the regular hexagon construction. Let label these points: A, B, C & D. The centre of the circle is at O.

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Adjust the compass so that its length is equal to AC. With the point at A, draw an arc crossing at C.

Without adjusting the compass and with the point at D, draw an arc crossing at B. These two arcs meet at E.

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Adjust the compass so that its length is equal to OE. The length of OE equals the side length of the square.

Starting at A, mark off 4 arc crossings around the circle. Joining these arc crossings with your straightedge completes the inscribed square construction.

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Regular Pentagon

Of all the inscribed polygons, the regular pentagon has far and away the most complex construction.

Draw a circle with centre O and diameter AB.

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Placing the compass point at B adjust the compass to length OB. Draw an arc that it passes through the centre O and cuts the circumference of the circle.

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Draw line between where the arc crosses the circle, labelling the intersection with the diameter as C.

C is the midpoint of OB.

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Bisect the diameter AB. D is the intersection of this bisector and the circle.

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Place the compass point at C and adjust its length to equal CD. Draw an arc so that crosses the diameter AB at point E.

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The length of line segment DE is equal to the side length of a regular pentagon inscribed in the circle.

With the compass point at D and adjusted to length DE, mark off 5 arcs around the circle.

Join the arc crossings to form an inscribed regular pentagon.

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