Euler's Formula for Polyhedra

Learn Euler's formula V - E + F = 2, count faces, edges and vertices systematically, and use the formula to find missing counts and test whether a solid can exist.

Advanced25 min18 problems
Practice Problems

Definition

A polyhedron is a solid whose surface is made only of flat polygons. Every polyhedron has three counts:
  • VV = number of vertices (corners)
  • EE = number of edges (segments where two faces meet)
  • FF = number of faces (the flat polygons)
Euler's formula says that for every convex polyhedron:
V−E+F=2V - E + F = 2
For a cube: V=8V = 8, E=12E = 12, F=6F = 6, and 8−12+6=28 - 12 + 6 = 2.
Counting edges systematically: every edge belongs to exactly two faces. So if you add up the number of sides of all faces, each edge is counted twice:
E=sum of the sides of all faces2E = \frac{\text{sum of the sides of all faces}}{2}
Prisms and pyramids with an nn-sided base follow simple patterns:
  • Prism: V=2nV = 2n, E=3nE = 3n, F=n+2F = n + 2
  • Pyramid: V=n+1V = n + 1, E=2nE = 2n, F=n+1F = n + 1
The formula only holds for solids without holes. A picture frame shaped solid (a solid with one hole through it) gives V−E+F=0V - E + F = 0 instead of 22.

Try it now

Which equation is Euler's formula for a convex polyhedron?

Worked Examples

Count the vertices, edges and faces of a hexagonal prism and check Euler's formula.
1

Count the vertices layer by layer

The top hexagon has 66 corners and the bottom hexagon has 66 corners → V=6+6=12V = 6 + 6 = 12
2

Count the edges in groups

66 edges around the top, 66 around the bottom, 66 vertical edges joining them → E=6+6+6=18E = 6 + 6 + 6 = 18
3

Count the faces

22 hexagonal bases and 66 rectangular side faces → F=2+6=8F = 2 + 6 = 8
4

Check Euler's formula

V−E+F=12−18+8V - E + F = 12 - 18 + 8 → 22

Common Mistakes

Adding up the sides of all faces and calling that the number of edges

Why it's wrong: Every edge is the meeting line of two faces, so it is counted once for each of them.

Correct: Divide the total number of face sides by 2. A cube has 6×4=246 \times 4 = 24 face sides but only 24÷2=1224 \div 2 = 12 edges.

Mixing up the signs, for example writing V+E−F=2V + E - F = 2

Why it's wrong: The edges are the count that is subtracted. With the wrong signs a cube gives 8+12−6=148 + 12 - 6 = 14.

Correct: Vertices and faces are added, edges are subtracted: V−E+F=2V - E + F = 2. Check it on a cube: 8−12+6=28 - 12 + 6 = 2.

Using the formula on cylinders, cones, spheres or solids with holes

Why it's wrong: Curved surfaces are not polygons, and a hole changes the result (a picture frame gives 00).

Correct: Use V−E+F=2V - E + F = 2 only for polyhedra without holes: flat polygon faces, straight edges, one closed surface.

Thinking that any three numbers with V−E+F=2V - E + F = 2 describe a real polyhedron

Why it's wrong: The formula is a test that every polyhedron passes, but passing it is not enough. For example V=4.5V = 4.5, E=7.5E = 7.5, F=5F = 5 also gives 22.

Correct: Also check that the counts are whole numbers and that the face sides add up to an even number (2E2E). Five triangles have 1515 sides, which is odd, so no polyhedron is made of exactly five triangles (a polyhedron always has an even number of odd-sided faces).

Interactive Visual

Count V, E and F: prisms, pyramids and an octahedron

Pick a solid in the table, then count its vertices (V), edges (E) and faces (F) below. Tap ? to reveal a number instead.

SolidVEFV − E + F
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Count them

Tap the solid or Start counting: every vertex lights up one by one.

Tap through the edges one by one

Count them

Tap the solid or Start counting: every edge lights up one by one.

Faces

8

Edges

18

Vertices

12

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which equation is Euler's formula for a convex polyhedron?

Why It Matters

Euler's formula connects three numbers that seem unrelated, and it works for every convex polyhedron, no matter its size or shape.
  • Checking your counts: if you count VV, EE and FF and do not get 22, you made a counting mistake (or the solid has a hole)
  • Finding a missing number: knowing two of the counts gives you the third without counting
  • Testing designs: engineers and 3D artists use it to check that a closed mesh has no gaps or holes
  • Sports and science: the panels of a classic soccer ball, the shapes of crystals and the frameworks of geodesic domes all obey it
It is also your first step into topology, the part of mathematics that studies what stays the same when shapes are stretched or bent.

Real World Applications

Designing a Soccer Ball

The classic soccer ball is stitched from $12$ pentagons and $20$ hexagons. Designers use face sides and Euler's formula to know how many seams and seam corners there are before cutting any material.

Example:

Each seam joins two panels, so the number of seams is half the number of panel sides.

1Try It Yourself

A soccer ball has 1212 pentagonal and 2020 hexagonal panels. Seams are the edges, the points where three seams meet are the vertices.

How many vertices does the ball have?

Step 1: Write the mathematical expression

Use V=2+E−FV = 2 + E - F with the number of seams EE and panels FF:

3D Models in Games and Animation

Characters and objects in video games are built from closed meshes of triangles. Software checks Euler's formula to catch meshes with gaps, duplicated corners or accidental holes before they cause glitches.

Example:

A closed mesh made of 44 triangles (a tetrahedron) has 66 edges and 44 vertices: 4−6+4=24 - 6 + 4 = 2.

2Try It Yourself

A 3D artist builds a closed model, with no holes, from 500500 triangles.

How many vertices should the model have?

Step 1: Write the mathematical expression

Find EE from the triangle sides, then V=2+E−FV = 2 + E - F:

Crystals in Minerals

Many crystals grow as polyhedra. A quartz crystal attached to a rock is shaped like a hexagonal prism with a hexagonal pyramid on its free end, and mineralogists use counts like these to describe crystal forms.

Example:

A quartz point has 66 rectangular sides, 66 triangular faces at the tip and 11 hexagonal face where it sits on the rock.

3Try It Yourself

A quartz crystal is a hexagonal prism with a hexagonal pyramid on top. It has 1313 faces and 2424 edges.

How many vertices does the crystal have?

Step 1: Write the mathematical expression

Solve V−E+F=2V - E + F = 2 for VV:

Key Takeaways

  • 1A polyhedron is a solid with flat polygon faces; count its vertices VV, edges EE and faces FF
  • 2Euler's formula: V−E+F=2V - E + F = 2 for every convex polyhedron (and every polyhedron without holes)
  • 3Count systematically in groups (top, bottom, sides) and check your counts with the formula
  • 4Every edge belongs to two faces, so EE = (sum of the sides of all faces) ÷2\div 2
  • 5Prism with an nn-gon base: V=2nV = 2n, E=3nE = 3n, F=n+2F = n + 2; pyramid: V=n+1V = n + 1, E=2nE = 2n, F=n+1F = n + 1
  • 6Counts that do not give 22 cannot belong to a convex polyhedron; a solid with one hole gives V−E+F=0V - E + F = 0

Frequently Asked Questions

Yes, as long as the solid has no holes (it could be inflated like a balloon into a sphere). An L-shaped block, for example, still gives V−E+F=2V - E + F = 2. Solids with holes, like a picture frame, give a different value.
Yes, as long as the solid has no holes (it could be inflated like a balloon into a sphere). An L-shaped block, for example, still gives V−E+F=2V - E + F = 2. Solids with holes, like a picture frame, give a different value.
Imagine removing one face and flattening the rest onto a table. Splitting the faces into triangles and then removing outer triangles one at a time never changes V−E+FV - E + F, and you end with a single triangle (3−3+1=13 - 3 + 1 = 1). Adding back the removed face gives 1+1=21 + 1 = 2. The number 22 belongs to the sphere-like shape of the surface, not to the particular solid.
No. They have curved surfaces, so they are not polyhedra and their faces and edges are not counted the same way. Euler's formula is about solids made only of flat polygons.
Leonhard Euler described it in a letter in 1750 and published it in the following years. René Descartes had written down a closely related result around 1630, but his notes were only found much later.

Glossary

Polyhedron
A solid whose surface is made only of flat polygons (plural: polyhedra)
Face
One of the flat polygons that make up the surface of a polyhedron
Edge
A line segment where two faces of a polyhedron meet
Vertex
A corner point where edges meet (plural: vertices)
Convex polyhedron
A polyhedron with no dents: the segment between any two of its points stays inside it
Euler's formula
The rule V−E+F=2V - E + F = 2 for every convex polyhedron

Formula Card

Euler's Formula

V−E+F=2V - E + F = 2

$V$ = vertices, $E$ = edges, $F$ = faces of a convex polyhedron

Edges from the Faces

E=sum of face sides2E = \frac{\text{sum of face sides}}{2}

Every edge is shared by exactly two faces

Prism with an n-gon Base

V=2n,  E=3n,  F=n+2V = 2n,\; E = 3n,\; F = n + 2

$n$ = number of sides of the base

Pyramid with an n-gon Base

V=n+1,  E=2n,  F=n+1V = n + 1,\; E = 2n,\; F = n + 1

$n$ = number of sides of the base

Today's challengeOne problem, three tries

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