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# 4. Platonic Solids

• • Contributed by Pamini Thangarajah
• Professor (Mathematics & Computing) at Mount Royal University
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Goal: To appreciate polygons and support the idea that there are exactly 5 platonic solids.

Terminology:

• A polygon is a closed 2-dimensional figure with straight sides

• An n-gon is a polygon with exactly n sides

• A regular n-gon is a polygon with exactly n sides, where all sides are of equal length and all interior angles of the polygon are equal. The sum of the interior angles of a regular n-gon is 180°(n - 2). It follows that each interior angle must measure 180°(n - 2)/n. So:

• A regular 3-gon is an equilateral triangle. Each interior angle is 60°

• A regular 4-gon is a square. Each interior angle is 90°

• A regular 5-gon is a regular pentagon. Each interior angle is 108°

• A regular 6-gon is a regular hexagon. Each interior angle is 120°

• A regular 7-gon is a regular heptagon. Each interior angle is 900/7°, or approximately 128.6°

• A regular 8-gon is a regular octagon. Each interior angle is 135°

Activity:

Suppose I want to tape regular n-gons together to make 3-dimensional shapes. I can make a cube, for example, by taping squares together. What are my options? I don’t want to bend or fold the n-gons. Let’s just concentrate on the corners of these objects.

Fact: To make a corner I’ll need at least 3 regular n-gons.

Try making corners out of 3 n-gons. Which ones will work? Justify your conclusions.

Now try using four n-gons to make corners. Which ones will work? Justify your conclusions.