Euler's Formula (Polyhedra)

Euler's Formula (Polyhedra)

(number of faces) + (number of vertices) – (number of edges) = 2

This formula is true for all convex polyhedra as well as many types of concave polyhedra.

Note: Euler is pronounced "Oiler".

Euler's Formula: faces + vertices - edges = 2. Example: A cube-like polyhedron with 7 faces, 10 vertices, 15 edges; 7+10-15=2.

Key Formula

F + V - E = 2

Where:

$F$ = Number of faces of the polyhedron
$V$ = Number of vertices (corners) of the polyhedron
$E$ = Number of edges of the polyhedron

Worked Example

Problem: Verify Euler's formula for a cube.

Step 1: Count the faces of a cube. A cube has 6 square faces (top, bottom, front, back, left, right).

F = 6

Step 2: Count the vertices. A cube has 8 corners.

V = 8

Step 3: Count the edges. A cube has 12 edges (4 on top, 4 on bottom, 4 vertical).

E = 12

Step 4: Substitute into Euler's formula and check that the result equals 2.

F + V - E = 6 + 8 - 12 = 2 \checkmark

Answer: The cube satisfies Euler's formula: 6 + 8 − 12 = 2.

Another Example

This example uses the formula to find a missing value rather than just verifying the formula, showing its practical problem-solving use.

Problem: A convex polyhedron has 5 faces and 6 vertices. Use Euler's formula to find the number of edges.

Step 1: Write down the known values.

F = 5, \quad V = 6

Step 2: Substitute into Euler's formula and solve for E.

5 + 6 - E = 2

Step 3: Simplify and isolate E.

11 - E = 2 \implies E = 9

Step 4: Check: this matches a triangular prism (2 triangular faces + 3 rectangular faces = 5 faces, 6 vertices, 9 edges).

5 + 6 - 9 = 2 \checkmark

Answer: The polyhedron has 9 edges.

Frequently Asked Questions

Does Euler's formula work for all polyhedra?

Euler's formula works for all convex polyhedra and for any polyhedron that is topologically equivalent to a sphere (meaning it has no holes through it). It does not work for polyhedra with holes, such as a torus-shaped solid. For a solid with one hole, the formula becomes F + V − E = 0 instead of 2.

What is Euler's formula for a tetrahedron?

A tetrahedron has 4 triangular faces, 4 vertices, and 6 edges. Substituting into Euler's formula gives 4 + 4 − 6 = 2, which confirms the formula holds. The tetrahedron is the simplest convex polyhedron.

Is Euler's formula for polyhedra the same as Euler's formula in trigonometry?

No. Euler's formula for polyhedra (F + V − E = 2) relates faces, vertices, and edges of 3D solids. Euler's formula in trigonometry and complex analysis states that e^(iθ) = cos θ + i sin θ. They are completely different results, both named after the Swiss mathematician Leonhard Euler.

Euler's Formula (Polyhedra) vs. Euler's Formula (Complex Numbers)

	Euler's Formula (Polyhedra)	Euler's Formula (Complex Numbers)
Formula	F + V − E = 2	e^(iθ) = cos θ + i sin θ
Branch of math	Geometry / Topology	Complex analysis / Trigonometry
What it relates	Faces, vertices, and edges of a 3D solid	Exponential and trigonometric functions via imaginary numbers
Typical grade level	Middle school through high school geometry	Advanced high school or college mathematics

Why It Matters

Euler's formula appears in geometry courses whenever you study 3D solids such as prisms, pyramids, and Platonic solids. It provides a quick way to find a missing count (faces, vertices, or edges) when the other two are known. Beyond the classroom, the formula is a foundational result in topology — the study of shapes and surfaces — and it underpins ideas in computer graphics, network theory, and architecture.

Common Mistakes

Mistake: Mixing up the signs and writing F + V + E = 2 or F − V + E = 2.

Correction: Remember the correct formula is F + V − E = 2. A helpful mnemonic: Faces and Vertices are added (both positive), while Edges are subtracted.

Mistake: Applying the formula to a shape with a hole (like a hollow torus) and expecting the result to equal 2.

Correction: Euler's formula equals 2 only for polyhedra without holes (genus 0). For a polyhedron with g holes, the general form is F + V − E = 2 − 2g. Always check that the solid has no tunnels or handles before assuming the result is 2.

Related Terms

Polyhedron — The 3D solid the formula describes
Face of a Polyhedron — The flat surfaces counted as F
Vertex — The corners counted as V
Edge of a Polyhedron — The line segments counted as E
Convex — Property guaranteeing the formula holds
Concave — Some concave polyhedra also satisfy the formula
Formula — General term for a mathematical relationship
Equation — Euler's formula is expressed as an equation