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Mathematics LibreTexts

4. Platonic Solids

  • Page ID
  • [ "stage:draft", "article:topic", "Platonic solids", "authorname:thangarajahp", "license:ccbysa", "showtoc:yes" ]

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    Goal: To appreciate polygons and support the idea that there are exactly 5 platonic solids.



    • 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°



    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.