4. Platonic Solids
- Page ID
- 13600
This page is a draft and is under active development.
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Goal: To appreciate polygons and support the idea that there are exactly 5 platonic solids.
Terminology:
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A polygon is a closed 2-dimensional figure with straight sides
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An n-gon is a polygon with exactly n sides
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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:
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A regular 3-gon is an equilateral triangle. Each interior angle is 60°
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A regular 4-gon is a square. Each interior angle is 90°
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A regular 5-gon is a regular pentagon. Each interior angle is 108°
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A regular 6-gon is a regular hexagon. Each interior angle is 120°
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A regular 7-gon is a regular heptagon. Each interior angle is 900/7°, or approximately 128.6°
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A regular 8-gon is a regular octagon. Each interior angle is 135°
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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.
What about using five n-gons? Justify your conclusions.
Can we make corners out of six or more n-gons? Justify your conclusions.
A platonic solid is a 3-dimensional object made by taping together regular n-gons in such a way that each corner is the same, and has the same number of n-gons around it. Using the data you’ve gathered, please complete the following statement:
I have found that there are ____________ ways to tape regular n-gons together to make the corners of a platonic solid. Therefore, there are at most __________ platonic solids.