Geometry of 3-D Objects: Faces, Vertices, Edges & Nets

Classifying 3-D objects by their faces, vertices and edges, the five Platonic solids, Euler's formula, the properties of spheres and cylinders, and how nets fold into solids — with drawn figures, worked examples and a quiz.

Every solid object can be described by three counts: how many flat surfaces it has, how many corners, and how many lines where surfaces meet. Getting this vocabulary exact is the whole of this section.

1Faces, vertices and edges

A face is a flat (or curved) surface of the solid.
A vertex is a corner — a point where edges meet. (Plural: vertices.)
An edge is the straight line where two faces meet.

Worked ExampleCount the faces, vertices and edges of a cube

1
Faces: 6 squares — top, bottom and four sides.
Count opposite pairs: 2 + 2 + 2.
2
Vertices: 8 corners.
4 on the top square, 4 on the bottom.
3
Edges: 12.
4 around the top, 4 around the bottom, 4 vertical.

2Euler's formula

For every solid with flat faces there is a remarkable relationship between the three counts:

F + V − E = 2

Worked ExampleCheck Euler's formula for a cube

1
F = 6, V = 8, E = 12.
From the counts above.
2
6 + 8 − 12 = 2 ✓
The formula holds — a quick way to check your counting.

💡

If your three counts do not satisfy F + V − E = 2, you have miscounted. Use it to check yourself before writing the answer down.

3The five Platonic solids

A Platonic solid has faces that are all identical regular polygons, with the same number of faces meeting at every vertex. Remarkably, there are only five of them.

Tetrahedron

F 4 · V 4 · E 6

Cube

F 6 · V 8 · E 12

Octahedron

F 8 · V 6 · E 12

Dodecahedron

F 12 · V 20 · E 30

Icosahedron

F 20 · V 12 · E 30

The five Platonic solids, rendered from their true 3-D coordinates. Every one satisfies F + V − E = 2.

Tetrahedron — 4 triangular faces, 4 vertices, 6 edges
Cube (hexahedron) — 6 square faces, 8 vertices, 12 edges
Octahedron — 8 triangular faces, 6 vertices, 12 edges
Dodecahedron — 12 pentagonal faces, 20 vertices, 30 edges
Icosahedron — 20 triangular faces, 12 vertices, 30 edges

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Every one of the five satisfies Euler's formula. Check the icosahedron: 20 + 12 − 30 = 2 ✓

4Spheres and cylinders

A sphere: every point on the surface is the same distance (the radius) from the centre.

A sphere has one curved surface, no edges and no vertices. Every surface point is the same distance from the centre.
A cylinder has two flat circular faces plus one curved surface, 2 curved edges and no vertices.

⚠️

Euler's formula F + V − E = 2 applies to solids with flat faces. Do not try to use it on a sphere or a cylinder.

5Nets

A net is the flat shape you get by unfolding a solid. Folding the net back up rebuilds the solid, so a net must contain exactly one piece for every face.

One net of a cube. A cube has 6 faces, so every cube net has exactly 6 squares.

Worked ExampleWhat solid does a net of 2 circles and 1 rectangle make?

1
Two circles become the flat top and bottom.
Circular faces.
2
The rectangle wraps around to form the curved surface.
Its length equals the circumference of the circle.
3
The solid is a cylinder.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
How many faces, vertices and edges does a cube have?
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6 faces, 8 vertices, 12 edges
2
State Euler's formula.
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F + V − E = 2
3
A solid has 5 faces and 6 vertices. How many edges?
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5 + 6 − E = 2 → E = 9
4
How many Platonic solids are there?
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Five
5
How many faces does an octahedron have, and what shape are they?
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8 triangular faces
6
Check Euler's formula for a tetrahedron (4 faces, 4 vertices, 6 edges).
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4 + 4 − 6 = 2 ✓
7
How many edges and vertices does a sphere have?
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None — a sphere has one curved surface only.
8
How many squares are in the net of a cube?
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6, one for each face
9
A net has 2 circles and 1 rectangle. Which solid is it?
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A cylinder