A polyhedron has 10 faces, 16 vertices and 24 edges, and these numbers fit Euler's formula. True or false?

True. Put them in F + V − E: 10 + 16 − 24 must equal 2. The edges that fit are 10 + 16 − 2 = 24.

A corner is cut off a prism on a 7-sided base, the cut a triangle. How many edges does the solid left have?

24 edges. Cutting the corner adds one face, the cut: 10; the one vertex becomes the cut's 3: 16; so E = F + V − 2 = 24.

3-D figures · Ages 12–13

Euler's formula and regular polyhedra

F + V − E = 2; edges = all the faces' sides ÷ 2.

How to do it

Method

F + V − E = 2; edges = all the faces' sides ÷ 2

A cube: 6 + 8 − 12 = 2

Another way

Count from the faces

Icosahedron: 20 × 3 ÷ 2 = 30 edges

In full

For every polyhedron without holes, the faces F, vertices V and edges E fit Euler's formula: F + V − E = 2. Know two and you can find the third: E = F + V − 2, V = 2 − F + E, F = 2 − V + E. There are only 5 regular polyhedra: the regular tetrahedron (4 equilateral triangles), the cube (6 squares), the regular octahedron (8 triangles), the regular dodecahedron (12 regular pentagons) and the regular icosahedron (20 triangles). Count edges and vertices from the faces: each edge belongs to 2 faces, so E = all the faces' sides ÷ 2; m faces meet at each vertex, so V = all the faces' sides ÷ m. The icosahedron: 20 × 3 ÷ 2 = 30 edges and 20 × 3 ÷ 5 = 12 vertices; check: 20 + 12 − 30 = 2. Why only 5? At least 3 faces meet at a vertex, and their angles must add to less than 360°: 3, 4 or 5 triangles, 3 squares or 3 pentagons; 3 hexagons already make 360° and lie flat.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
How many edges does the regular polyhedron in the picture have?
Show the answer and working
Answer: 12 edges
It has 6 faces of 4 sides each, and each edge belongs to two faces: 6 × 4 ÷ 2 = 12. Or count them, dashed ones too.
Example 2
A polyhedron has 10 faces, 16 vertices and 24 edges, and these numbers fit Euler's formula.
True or false?
Show the answer and working
Answer: True
Put them in F + V − E: 10 + 16 − 24 must equal 2. The edges that fit are 10 + 16 − 2 = 24.
Example 3
A corner is cut off a prism on a 7-sided base, the cut a triangle. How many edges does the solid left have?
Show the answer and working
Answer: 24 edges
Cutting the corner adds one face, the cut: 10; the one vertex becomes the cut's 3: 16; so E = F + V − 2 = 24.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.A polyhedron has 5 faces and 5 vertices. How many edges does it have?
Answer:
2.A polyhedron has 6 vertices and 9 edges. How many faces does it have?
Answer:
3.A polyhedron has 8 faces, 8 vertices and 16 edges, and these numbers fit Euler's formula.
True or false?
Answer:
4.6 equal regular polygons with 5 sides each meet at every vertex. This makes a regular polyhedron.
True or false?
Answer:
5.A corner is cut off a prism on a 5-sided base, the cut a triangle. How many edges does the solid left have?
Answer:

All the answers

1. 8 edges
2. 5 faces
3. False
4. False
5. 18 edges

Free worksheet: Euler's formula and regular polyhedraWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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