Polyhedron

A polyhedron is a three-dimensional solid whose surface is made entirely of flat polygons. The polygons are its faces, the straight lines where two faces meet are its edges, and the corners where edges meet are its vertices. Cubes, pyramids and prisms are polyhedra. Spheres, cylinders and cones are not, because they have curved surfaces. The plural is polyhedra or polyhedrons.

What is a polyhedron?

The word comes from Greek: poly means many and hedron refers to a face. To be a polyhedron, a solid needs:

• Flat faces that are polygons: triangles, squares, pentagons and so on.
• Straight edges.
• No curved surfaces and no gaps: the faces close up completely around the inside.

Faces, edges and vertices

To count edges without missing any, count them by layer: a cube has 4 around the top, 4 around the bottom and 4 going up the sides, making 12.

Euler's formula

For every convex polyhedron, and for any polyhedron without holes through it, the numbers of vertices (V), edges (E) and faces (F) are linked by Euler's formula:

For a cube, 8 − 12 + 6 = 2. It works for every shape in the tables below. A polyhedron with a hole through it, like a picture frame shape, gives a different answer, which is how mathematicians tell such shapes apart.

Example 1: find the number of edges

A polyhedron has 8 vertices and 12 faces. How many edges does it have?

It has 18 edges. (A hexagonal bipyramid, two hexagonal pyramids joined base to base, fits these numbers.)

Example 2: can this shape exist?

Could a convex polyhedron have 6 vertices, 10 edges and 5 faces? Check: 6 − 10 + 5 = 1, not 2. So no: someone has miscounted.

Types of polyhedra

Prisms

A prism has two identical, parallel polygon faces, called the bases, joined by rectangles (in a right prism). It is named after its base: a triangular prism, a rectangular prism (a box), a pentagonal prism and so on. A cube is a rectangular prism with square faces.

Pyramids

A pyramid has one polygon base, and triangles that rise from each edge of the base to meet at a single point, the apex. It is named after its base: a square pyramid, like those in Egypt, has a square base and 4 triangles. A triangular pyramid is also called a tetrahedron.

Counting faces, edges and vertices of prisms and pyramids

Each row passes Euler's test: for a pentagonal prism, 10 − 15 + 7 = 2.

The 5 Platonic solids

A regular polyhedron, or Platonic solid, has faces that are all the same regular polygon, with the same number of faces meeting at every vertex. There are exactly five, a fact known since ancient Greece.

Board game players know them as dice: the d4, d6, d8, d12 and d20 are the five Platonic solids. Why only five? At each vertex, at least three faces must meet, and their angles must add up to less than 360°, or the surface would lie flat. Only triangles (three, four or five at a vertex), squares (three) and pentagons (three) fit that rule.

Nets of polyhedra

A net is the flat pattern you get by unfolding a polyhedron along some of its edges. Fold it back up and you get the solid. A cube has 11 different nets. Here is one:

Nets are useful for surface area: the total area of the net is the surface area of the solid. Under the Common Core standards, grade 6 students use nets made of rectangles and triangles to find surface areas.

Check your understanding.

Which shapes are not polyhedra?

Convex and concave polyhedra

A polyhedron is convex if a straight line joining any two points inside it stays inside it: no dents. All the shapes above are convex. A concave polyhedron has a dent or a notch, like an L-shaped block. Concave polyhedra without holes still obey Euler's formula.

Common mistakes

• Calling a cylinder or cone a polyhedron. Any curved surface rules it out.
• Missing hidden edges and vertices at the back of a drawing. Count by layers.
• Mixing up a prism and a pyramid. A prism has two matching bases. A pyramid has one base and an apex.
• Naming by the side faces. Prisms and pyramids are named by the shape of their base.
• Getting Euler's formula backward. It is V − E + F = 2: vertices minus edges plus faces.

Practice with a short timed paper

Solids, nets, surface area and volume come up in middle school and high school geometry. The daily Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math papers are short timed practice tests of 8 questions, with a new one every day. For flat shapes, see what a polygon is.

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