How to prove that there can only be five Platonic solids

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A one-minute video on how to prove there are only five Platonic solids. In a Platonic solid every face is the same regular polygon: a triangle, square or pentagon. You can’t make a corner from regular hexagons as three together make a flat sheet.

How to prove that there can only be five Platonic solids slideshow

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Click to see a step-by-step slideshow.
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WHAT YOU NEED: Tetrahedron net, cube net, octahedron net, dodecahedron net, icosahedron net, scissors, glue stick and coloured pens
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STEP 1: On each of the nets, colour in the faces that will make one corner so you can count them easily.
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STEP 2: Fold and stick the nets together to form five different solids.
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STEP 3: Three regular triangles on a corner make a tetrahedron, four make an octahedron and five make an icosahedron. Six triangles make a flat sheet that can't be folded into a corner. So only three Platonic solids can be made from triangles.
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Three squares make the corner of a cube. However four squares just make a flat sheet that can’t be folded into a corner. Therefore only one Platonic solid can be made from squares: a cube.
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Three regular pentagons make the corner of a dodecahedron. But there isn’t room to fit any more than 3 pentagons together to make a corner. So only one Platonic solid can be made from pentagons.
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STEP 4: Three regular hexagons just make a flat sheet. And shapes with more sides, like heptagons or octagons, can’t fit together to make the minimum three faces to make a corner. Therefore we can only make five Platonic solids.
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These solids were named after the ancient Greek mathematician Plato. He believed the universe was made of five elements – fire, earth, water, air and spirit – each represented by one of these five solids.

How to prove that there can only be five Platonic solids slideshow

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