A Platonic solid (regular polyhedron) is a convex solid where every face is a congruent regular polygon and the same number of faces meet at every vertex. You might expect infinitely many such shapes, but there are actually exactly 5. See why for yourself by adjusting the angles.
A regular polygon on a flat plane can be made with any number of sides, without limit — but a 3D Platonic solid comes in only 5 varieties: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Why is the count fixed at exactly this number?
The secret lies in the total angle that gathers at a vertex. To fold paper into a point, the sum of the angles meeting at that one point has to be less than 360°. Exactly 360° unfolds flat, and anything more can't be folded at all because the paper would overlap itself. No matter how you combine the number of sides of a regular polygon (n) with the number of faces meeting at a vertex (p), only 5 combinations satisfy this condition.
Use the sliders to change n and p, and see for yourself the exact moment the shape stops folding once the angle sum reaches or passes 360°.
Several regular polygons gather at a Platonic solid's vertex. The sum of the gathered angles has to be less than 360° for it to fold into a pointed 3D corner (vertex), just like paper. Equal to 360° and it unfolds flat; more than 360° and it can't fold at all because it overlaps. Try changing n (number of sides) and p (number of faces meeting at a vertex) to see for yourself.
A regular polygon's interior angle grows larger as it gets more sides (equilateral triangle 60°, square 90°, regular pentagon 108°, regular hexagon 120°...). And at least 3 faces have to meet at a vertex for a solid to form at all. Apply these two conditions together with the rule "angle sum < 360°," and only the following 5 combinations hold up.
Pick a shape below, and freely rotate it by dragging with your mouse or finger.
| Name | Schläfli symbol | Face shape | Faces at a vertex | V | E | F |
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