4. Platonic Solids
 Page ID
 13600
Goal: To appreciate polygons and support the idea that there are exactly 5 platonic solids.
Terminology:

A polygon is a closed 2dimensional figure with straight sides

An ngon is a polygon with exactly n sides

A regular ngon 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 ngon is 180°(n  2). It follows that each interior angle must measure 180°(n  2)/n. So:

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

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

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

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

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

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


Activity:
Suppose I want to tape regular ngons together to make 3dimensional 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 ngons. Let’s just concentrate on the corners of these objects.
Fact: To make a corner I’ll need at least 3 regular ngons.
Try making corners out of 3 ngons. Which ones will work? Justify your conclusions.
Now try using four ngons to make corners. Which ones will work? Justify your conclusions.
What about using five ngons? Justify your conclusions.
Can we make corners out of six or more ngons? Justify your conclusions.
A platonic solid is a 3dimensional object made by taping together regular ngons in such a way that each corner is the same, and has the same number of ngons around it. Using the data you’ve gathered, please complete the following statement:
I have found that there are ____________ ways to tape regular ngons together to make the corners of a platonic solid. Therefore, there are at most __________ platonic solids.