What Is a Polyhedron? Types & Euler's Formula

TL;DR

A polyhedron is a three-dimensional solid bounded entirely by flat polygon faces, joined along straight edges that meet at points called vertices. This article defines the polyhedron, names its parts, walks through the types, and shows how Euler's formula, F+V−E=2, ties faces, vertices, and edges together.

Last updated on June 15, 2022, 9 min read

What Is a Polyhedron?

A polyhedron is a closed three-dimensional solid whose surface is made up entirely of flat polygon faces. The word comes from the Greek poly ("many") and hedron ("face") — literally "many faces." Its surface has no curves and no openings: every part of the boundary is a flat polygon.

Every polyhedron has three kinds of parts:

• Faces — the flat polygon surfaces that bound the solid (a cube has 6 square faces).
• Edges — the straight line segments where two faces meet (a cube has 12 edges).
• Vertices — the corner points where three or more edges meet (a cube has 8 vertices).

A reader question worth settling immediately — is a sphere a polyhedron? No. A sphere, a cylinder, and a cone all have at least one curved surface, so none of them is a polyhedron; a polyhedron's faces must be flat polygons, its edges straight, its corners sharp. The smallest possible polyhedron is the tetrahedron, with just 4 triangular faces.

Types of Polyhedrons

Polyhedra split into families by the shape and arrangement of their faces.

• Prisms — two identical, parallel polygon faces (the bases) joined by rectangular side faces. A rectangular box and a triangular prism are prisms; the cross-section is the same all along its length.
• Pyramids — one polygon base and triangular side faces rising to a single apex point. A square pyramid (the Egyptian shape) and a triangular pyramid are the common ones.
• Platonic solids — the five regular polyhedra, where every face is the same regular polygon and the same number meet at each vertex: the tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), and icosahedron (20 triangles). There are exactly five — no more are geometrically possible.

Polyhedra are also sorted two more ways:

• Regular vs irregular. A regular polyhedron has identical regular-polygon faces; an irregular polyhedron does not.
• Convex vs concave. A convex polyhedron has no "dents" — any line segment joining two points inside it stays inside. A concave polyhedron has at least one inward dent. Euler's formula below holds for any convex polyhedron.

Euler's Formula: F+V−E=2

Here is the result that ties the three parts of a polyhedron together. For any convex polyhedron, the number of faces F, vertices V, and edges E obey: F+V−E=2.

Read it as a balance: faces plus vertices always exceed edges by exactly 2, no matter the shape or size of the convex polyhedron. Each letter is one of the parts you just met — F counts the flat faces, V the corner points, E the line segments between them.

Test it on a cube. A cube has F=6 faces, V=8 vertices, and E=12 edges: F+V−E=6+8−12=2.

Now on a triangular prism: F=5 (two triangles plus three rectangles), V=6, E=9: 5+6−9=2.

The formula does double duty. As a check, it confirms your part-counts are right. As a tool, it finds a missing count from the other two — rearrange it to E=F+V−2, V=E−F+2, or F=E−V+2.

Examples of Polyhedrons

With the parts, the types, and Euler's formula in place, here is the topic doing real work.

Example 1

How many faces, edges, and vertices does a square pyramid have?
A square pyramid has a square base and four triangular sides: F=5 faces, E=8 edges, V=5 vertices.

Example 2

A solid has 6 faces, 8 vertices, and a curved top. Is that a polyhedron?
No. A polyhedron's surface must be made entirely of flat polygon faces.

Example 3

A polyhedron has F=7 faces and V=10 vertices. Find the number of edges.
Use Euler's formula rearranged for edges: E=F+V−2=7+10−2=15 edges.

Example 4

Verify Euler's formula for an octahedron, which has 8 faces and 6 vertices and 12 edges.
F+V−E=8+6−12=2.✓

Example 5

A prism has a hexagonal base. How many faces, edges, and vertices does it have, and does it satisfy Euler's formula?
A hexagonal prism has 8 faces; it has 12 vertices and 18 edges. Check: 8+12−18=2.

Example 6

A convex polyhedron has 12 edges and 6 vertices. How many faces does it have?
F=E−V+2=12−6+2=8 faces.

Why Polyhedrons Matter Beyond the Classroom

Polyhedra are important in nature and engineering, building strong, packable, predictable solids out of flat pieces.

Key Takeaways

• A polyhedron is a 3D solid bounded entirely by flat polygon faces, with straight edges and corner vertices.
• Curved solids — spheres, cylinders, cones — are not polyhedra.
• The main types are prisms, pyramids, and the five Platonic solids.
• Euler's formula, F+V−E=2, links faces, vertices, and edges for any convex polyhedron and finds a missing count from the other two.
• The most common mistake is calling a curved solid a polyhedron or miscounting hidden edges.