Polyhedra & Platonic Solids Cheat Sheet

A printable reference covering polyhedra, Euler’s formula, prisms, pyramids, surface area, volume, and Platonic solids for grades 7-11.

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This cheat sheet covers the main ideas students need for working with polyhedra and Platonic solids. It explains how faces, edges, and vertices fit together in three-dimensional shapes. Students use these ideas to classify solids, check diagrams, and solve geometry problems involving surface area and volume. It is especially useful when moving from two-dimensional polygons to three-dimensional geometry. The core relationship for many polyhedra is Euler’s formula, $V - E + F = 2$ , where $V$ is vertices, $E$ is edges, and $F$ is faces. Prisms and pyramids have predictable volume formulas, such as $V = Bh$ for prisms and $V = \frac{1}{3}Bh$ for pyramids. Regular polyhedra have congruent regular polygon faces, and the five Platonic solids are the only convex regular polyhedra. Understanding nets, cross-sections, and symmetry helps connect visual models to formulas.

Key Facts

Euler’s formula for any convex polyhedron is $V - E + F = 2$ , where $V$ is vertices, $E$ is edges, and $F$ is faces.
The volume of a prism is $V = Bh$ , where $B$ is the area of the base and $h$ is the perpendicular height.
The volume of a pyramid is $V = \frac{1}{3}Bh$ , because a pyramid with the same base and height as a prism has one third the volume.
The surface area of a polyhedron is the sum of the areas of all its faces, written as $SA = A_1 + A_2 + A_3 + \cdots + A_n$ .
A regular polyhedron has faces that are congruent regular polygons, with the same number of faces meeting at every vertex.
There are exactly five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
For a prism with an $n$ -sided base, the numbers of vertices, edges, and faces are $V = 2n$ , $E = 3n$ , and $F = n + 2$ .
For a pyramid with an $n$ -sided base, the numbers of vertices, edges, and faces are $V = n + 1$ , $E = 2n$ , and $F = n + 1$ .

Vocabulary

Polyhedron: A three-dimensional solid made only of flat polygon faces, straight edges, and vertices.
Face: A flat polygon surface that forms part of the boundary of a polyhedron.
Edge: A line segment where two faces of a polyhedron meet.
Vertex: A point where three or more edges of a polyhedron meet.
Platonic solid: A convex regular polyhedron whose faces are congruent regular polygons and whose vertices are all arranged the same way.
Net: A two-dimensional pattern of polygons that can be folded to form a three-dimensional solid.

Common Mistakes to Avoid

Counting hidden edges incorrectly, because a drawing of a three-dimensional solid may not show every edge clearly. Use the structure of the solid or a formula such as $V - E + F = 2$ to check the count.
Using slant height as vertical height in volume formulas, because $V = Bh$ and $V = \frac{1}{3}Bh$ require perpendicular height. Slant height is used in many surface area problems, not in basic volume formulas.
Forgetting to include both bases in prism surface area, because a prism has two congruent base faces. Add the lateral area and both base areas, often written as $SA = Ph + 2B$ for right prisms.
Assuming every solid with regular polygon faces is a Platonic solid, because the same number of faces must meet at every vertex. A solid can have regular faces but still fail to be one of the five Platonic solids.
Mixing up prisms and pyramids, because both are named by their base shape. A prism has two congruent parallel bases, while a pyramid has one base and triangular faces that meet at one apex.

Practice Questions

1 A convex polyhedron has $V = 12$ vertices and $F = 8$ faces. Use $V - E + F = 2$ to find $E$ .
2 A right rectangular prism has length $8\text{ cm}$ , width $5\text{ cm}$ , and height $6\text{ cm}$ . Find its volume using $V = Bh$ .
3 A square pyramid has base side length $10\text{ m}$ and perpendicular height $9\text{ m}$ . Find its volume using $V = \frac{1}{3}Bh$ .
4 Explain why a cube is a Platonic solid but a rectangular prism with unequal side lengths is not.

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