Maths
What are faces, edges and vertices, and how do you count them on 3D shapes?
Quick answer
Faces are the flat surfaces of a 3D shape, edges are the lines where two faces meet, and vertices are the corners. A cube has 6 faces, 12 edges and 8 vertices.
In one sentence
A face is a flat surface of a 3D shape, an edge is a line where two faces meet, and a vertex is a corner where edges meet.
Explained
Pick up a dice or a cardboard box. The flat sides you can put your hand on are the faces. Run your finger along a fold where two faces meet: that is an edge. Stop at a pointy corner: that is a vertex. One corner is a vertex; two or more are vertices.
box
tent
ice cream
The faces of a solid are 2D shapes: a cube's faces are squares, and a triangular prism has 2 triangles and 3 rectangles. Unfold the shape and you see all the faces at once in its net.
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid (rectangular prism) | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Pentagonal prism | 7 | 15 | 10 |
| Hexagonal prism | 8 | 18 | 12 |
| Triangular pyramid (tetrahedron) | 4 | 6 | 4 |
| Square-based pyramid | 5 | 8 | 5 |
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 |
| Sphere | 1 curved | 0 | 0 |
You do not have to memorise the table. Prisms and pyramids follow patterns. If the base has n sides:
- Prism: faces = n + 2, edges = 3n, vertices = 2n. (Two bases, n rectangles; n edges round each base and n going across.)
- Pyramid: faces = n + 1, edges = 2n, vertices = n + 1. (One base, n triangles; n edges round the base and n up to the top.)
There is also a famous check for any solid with flat faces and no holes (a polyhedron). It is Euler's formula:
F + V − E = 2 (faces + vertices − edges = 2)
For a cube: 6 + 8 − 12 = 2. If you count a shape and do not get 2, you have missed something.
Curved shapes are trickier, and books do not all agree. Most primary schools say a cylinder has 3 faces (2 flat circles and 1 curved surface), 2 curved edges and no vertices, and a cone has 2 faces, 1 edge and 1 vertex (its point, or apex). Some books say "faces" must be flat and call the curved part a surface instead. Euler's formula only works for flat-faced shapes. More about solids is on the 3D shapes page.
Examples
- Square-based pyramid: faces 1 square + 4 triangles = 5; edges 4 round the base + 4 up to the top = 8; vertices 4 + 1 = 5. Check: 5 + 5 − 8 = 2.
- Hexagonal prism (n = 6): faces 6 + 2 = 8, edges 3 × 6 = 18, vertices 2 × 6 = 12. Check: 8 + 12 − 18 = 2.
- Octahedron (two square-based pyramids stuck together): 8 faces, 12 edges, 6 vertices. Check: 8 + 6 − 12 = 2.
Common mistakes
- Missing the hidden back. You can only see 3 faces of a cube in a picture, but it has 6. Fix: count in groups (top, bottom, sides) or hold a real object.
- Counting an edge twice. Each edge belongs to two faces. Fix: count round the top, round the bottom, then the edges going up.
- Mixing up edges and vertices. Edges are lines; vertices are points. A cube has more edges (12) than vertices (8).
- Using Euler's formula on a cylinder or cone. It is only for flat-faced shapes with no holes.
Step by step
Step by step
- Name the shapeIs it a prism, a pyramid or a curved shape? Find the base and count its sides (n).
- Count the facesBases plus side faces: n + 2 for a prism, n + 1 for a pyramid.
- Count the edgesGo round each base, then count the edges joining them or rising to the top.
- Count the verticesCount the corners of each base, plus the top point for a pyramid.
- Check with EulerFaces + vertices − edges should equal 2.
Practice questions
Faces, edges and vertices
Try each one, then check your answer.
- How many faces does a cuboid have?
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6 - How many vertices does a cube have?
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4 on top + 4 on the bottom = 8 - How many edges does a square-based pyramid have?
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4 round the base + 4 to the top = 8 - Give the faces, edges and vertices of a triangular pyramid.
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4 faces, 6 edges, 4 vertices - Give the faces, edges and vertices of a triangular prism.
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5 faces, 9 edges, 6 vertices - Give the faces, edges and vertices of a pentagonal prism.
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n = 5: 5 + 2, 3 × 5, 2 × 5 = 7 faces, 15 edges, 10 vertices - Give the faces, edges and vertices of a hexagonal pyramid.
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n = 6: 6 + 1, 2 × 6, 6 + 1 = 7 faces, 12 edges, 7 vertices - Give the faces, edges and vertices of an octagonal prism.
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n = 8: 10 faces, 24 edges, 16 vertices - A pyramid has a base with 10 sides. Give its faces, edges and vertices.
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11 faces, 20 edges, 11 vertices - Check Euler's formula for a hexagonal prism.
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8 + 12 − 18 = 2, so it works. - A polyhedron has 6 faces and 12 edges. How many vertices?
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V = 2 − 6 + 12 = 8 - A polyhedron has 12 vertices and 30 edges. How many faces?
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F = 2 + 30 − 12 = 20 - A prism has 18 edges. What shape is its base?
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3n = 18, so n = 6: a hexagon (a hexagonal prism). - A pyramid has 9 faces. How many edges does it have?
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n + 1 = 9, so n = 8 and edges = 2 × 8 = 16 - Using the usual school rule, give the faces, edges and vertices of a cylinder.
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3 faces (2 flat, 1 curved), 2 edges, 0 vertices - Someone says a polyhedron has 7 faces, 10 vertices and 14 edges. Can that be right?
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7 + 10 − 14 = 3, not 2, so no.
Word problems
Faces, edges and vertices in real life
Try each one, then check your answer.
- You build a cube frame from straws and balls of modelling clay. How many straws and how many balls do you need?
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One straw per edge and one ball per vertex: 12 straws and 8 balls - You build a square-based pyramid frame from 10 cm straws. What length of straw do you use in total?
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8 edges × 10 cm = 80 cm - You paint every face of a triangular prism a different colour. How many colours do you need?
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5 - A box is 30 cm long, 20 cm wide and 10 cm tall. You stick tape along every edge. How much tape do you need?
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4 × (30 + 20 + 10) = 4 × 60 = 240 cm - A classic football is stitched from 32 panels and has 60 corners. How many seams (edges) does it have?
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E = F + V − 2 = 32 + 60 − 2 = 90
Questions people ask
What is the difference between a face, an edge and a vertex?
A face is a flat surface, an edge is the line where two faces meet, and a vertex is a corner point where edges meet.
How do you find the faces, edges and vertices of a prism?
Count the sides of its base, n. A prism has n + 2 faces, 3n edges and 2n vertices. A triangular prism (n = 3) has 5 faces, 9 edges and 6 vertices.
How do you find the faces, edges and vertices of a pyramid?
With n sides on the base, a pyramid has n + 1 faces, 2n edges and n + 1 vertices. A square-based pyramid has 5 faces, 8 edges and 5 vertices.
How many faces, edges and vertices does a cylinder have?
At school a cylinder usually has 3 faces (2 flat circles and 1 curved surface), 2 curved edges and 0 vertices.
How many faces, edges and vertices does a cone have?
Usually 2 faces (1 flat circle and 1 curved surface), 1 curved edge and 1 vertex, the point at the top (the apex).
Does a sphere have faces, edges or vertices?
A sphere has one curved surface and no edges or vertices.
Is there a formula for faces, edges and vertices?
Yes: Euler's formula, F + V − E = 2, works for every solid with flat faces and no holes. For a cube, 6 + 8 − 12 = 2.
Quick quiz
Faces, edges and vertices
Check what you know with 4 quick questions.
Question 1 of 4
How many edges does a cube have?
Practise it
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Practise problems on 3D solids
3D shape worksheets
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Related topics
3D shapes
3D shapes are solid shapes with length, width and height, like cubes, spheres, cylinders and pyramids. They have faces, edges and vertices.
2D shapes
2D shapes are flat shapes with length and width but no thickness, like circles, triangles, squares and hexagons. You can draw them on paper.
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
Surface area
Surface area is the total area of all the outside faces of a 3D shape. For a cube with side a, it is 6a², because a cube has six equal square faces.
Nets of 3D shapes
A net is a flat 2D pattern that folds up into a 3D shape. A cube's net is 6 squares joined edge to edge, and a cube has 11 different nets.
Symmetry
A shape has symmetry when one half is a mirror image of the other (line symmetry), or when it looks the same after being turned (rotational symmetry).