MathsICSE Class 8 · Representing 3-D in 2-D

Euler's Relation F + V − E = 2 and Nets of Solids (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

For a solid with flat faces, $F + V - E = 2$, where $F$ is the number of faces, $V$ the vertices and $E$ the edges. A prism with an $n$-sided base has $F = n + 2$, $V = 2n$, $E = 3n$; a pyramid with an $n$-sided base has $F = n + 1$, $V = n + 1$, $E = 2n$. A net folds into a solid, so its area equals the surface area.

Understand Euler's Relation and Nets of Solids

The CISCE curriculum asks you to count vertices, edges and faces and to verify Euler's relation for solids with flat faces: cubes, cuboids, tetrahedrons, prisms and pyramids.

Terms. A face is a flat surface, an edge is where two faces meet, a vertex (plural vertices) is a corner where edges meet.

Euler's relation.
$ $F + V - E = 2.$ $

Counts for common solids.

Solid	F	V	E
Cube / cuboid	6	8	12
Tetrahedron	4	4	6
Square pyramid	5	5	8
Triangular prism	5	6	9
Hexagonal pyramid	7	7	12

General families. Prism with an $n$ -gon base: $F = n + 2$ , $V = 2 n$ , $E = 3 n$ . Pyramid with an $n$ -gon base: $F = n + 1$ , $V = n + 1$ , $E = 2 n$ . Both satisfy Euler's relation: $(n + 2) + 2 n - 3 n = 2$ and $(n + 1) + (n + 1) - 2 n = 2$ .

Finding a missing number. Rearrange: $E = F + V - 2$ , $V = E - F + 2$ , $F = E - V + 2$ .

Nets. A net is a flat pattern that folds into a solid. A cube's net has six squares; a cuboid's net has three pairs of equal rectangles; a triangular prism's net has two triangles and three rectangles. The area of a net equals the total surface area of the solid. The curriculum limits joined or nested shapes to at most two.

Euler's Relation and Nets of Solids: formulas to remember

Euler's relation: $F + V - E = 2$; Solids with flat faces.
Prism with n-gon base: $F = n + 2, V = 2 n, E = 3 n$; Cube/cuboid: n = 4.
Pyramid with n-gon base: $F = n + 1, V = n + 1, E = 2 n$; Tetrahedron: n = 3.
Net area: $area of net = total surface area$; Cuboid: $2 (l b + bh + h l)$ .

How to solve Euler's Relation and Nets of Solids questions

Count F, V and E carefully, then test or use F + V - E = 2.

1Sketch the solid and count faces, vertices and edges separately.
2Compute $F + V - E$ and compare with $2$ .
3For a missing value rearrange Euler's relation.
4For a prism or pyramid, use the $n$ -gon formulas to check your counts.
5For a net, add the areas of all its faces.

Euler's Relation and Nets of Solids in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Paper model of a cuboid box

Suppose a gift box is a cuboid 5 cm long, 3 cm wide and 2 cm high, made from one flat sheet (its net).

Find the area of the net and verify Euler's relation for the box.

Try it first, then show the solutionSolution

Net area $= 2 (5 \times 3 + 3 \times 2 + 2 \times 5) = 2 (15 + 6 + 10) = 62$ cm $^{2}$ .
$F + V - E = 6 + 8 - 12 = 2$ .

Answer: 62 cm $^{2}$ ; Euler holds.

Memory hook: A net is the solid unfolded, so its area is the surface area.

Tricks to remember Euler's Relation and Nets of Solids

Short memory aids. Say each one aloud once, then use it on a question the same day.

FVE rule
Faces plus vertices is always two more than edges.
Use it when: Finding a missing count.
Prism or pyramid
Prism: 2 x n vertices, 3 x n edges. Pyramid: n + 1 of each of F and V.
Use it when: Quick counts.

Euler's Relation and Nets of Solids: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Edges of a cube · 1 mark · MCQBoard level

The number of edges of a cube is:
(a) 6 (b) 8 (c) 12 (d) 16

Try it first, then show the solutionSolution

4 on the top, 4 on the bottom, 4 vertical = 12.

Answer: (c)

Example 2: Verify for a prism · 3 marks · Short answerBoard level

A triangular prism has 5 faces, 6 vertices and 9 edges. Verify Euler's relation.

Try it first, then show the solutionSolution

$F + V = 5 + 6 = 11$ .
$F + V - E = 11 - 9 = 2$ .
Euler's relation holds.

Answer: Verified: 2

Example 3: Hexagonal pyramid · 3 marks · Short answerHarder board level

A pyramid has a hexagonal base. Find its faces, vertices and edges and verify Euler's relation.

Try it first, then show the solutionSolution

Faces: 6 triangular + 1 base $= 7$ . Vertices: 6 on the base + apex $= 7$ .
Edges: 6 on the base + 6 slanting $= 12$ .
$7 + 7 - 12 = 2$ .

Answer: F = 7, V = 7, E = 12; verified

Example 4: Missing value · 4 marks · Short answerHarder board level

A solid has 8 faces and 12 edges. Find its number of vertices, and name a solid with these numbers.

Try it first, then show the solutionSolution

$V = E - F + 2 = 12 - 8 + 2 = 6$ .
A regular octahedron has 8 triangular faces, 12 edges and 6 vertices.
Check: $8 + 6 - 12 = 2$ .

Answer: V = 6 (an octahedron)

Example 5: A prism with 24 edges · 4 marks · Long answerHOTS / competency

A prism has an $n$ -sided base and 24 edges. Find $n$ , and its numbers of faces and vertices. Verify Euler's relation.

Try it first, then show the solutionSolution

Edges $= 3 n = 24 \Rightarrow n = 8$ (an octagonal prism).
Faces $= n + 2 = 10$ ; vertices $= 2 n = 16$ .
$F + V - E = 10 + 16 - 24 = 2$ .

Answer: n = 8; F = 10, V = 16

Where students lose marks in Euler's Relation and Nets of Solids

The slips examiners see most often, and what to write instead.

Mistake: Counting the base twice or missing it
A pyramid has its base plus the triangular faces.
Wrong working
Hexagonal pyramid: 6 faces.
Right working
$6 + 1 = 7$ faces.
Do this: Count slanting faces and the base separately.
Usually costs: Verification mark.
Mistake: Using F + V + E or F - V + E
The relation is F + V - E = 2.
Wrong working
$6 - 8 + 12 = 10$ for a cube.
Right working
$6 + 8 - 12 = 2$ .
Do this: Remember: faces and vertices add, edges subtract.
Usually costs: Whole question.

Exam tips

Make a small table of F, V, E for each solid.
State the relation before substituting.
For nets, name the shapes (rectangles, triangles) and add their areas.

Practice questions on Euler's Relation and Nets of Solids

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The number of edges of a tetrahedron is:
(a) 4 (b) 6 (c) 8 (d) 12
Hint
Six edges: 3 around the base and 3 to the apex.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
A square pyramid has 5 faces, 5 vertices and 8 edges. Verify Euler's relation.
Hint
F + V - E.
Answer
$5 + 5 - 8 = 2$ . Verified.
Q3 · 3 marks · Short answerHarder board level
A solid has 7 faces and 10 vertices. Find its edges and name a prism that fits.
Hint
E = F + V - 2.
Answer
$E = 7 + 10 - 2 = 15$ . A pentagonal prism has $7$ faces, $10$ vertices and $15$ edges.
Q4 · 3 marks · Short answerHarder board level
A pyramid has an $n$ -sided base and 20 edges. Find $n$ , its faces and vertices, and check Euler's relation.
Hint
Edges = 2n.
Answer
$2 n = 20 \Rightarrow n = 10$ . Faces $= 11$ , vertices $= 11$ . $11 + 11 - 20 = 2$ .
Q5 · 4 marks · Case-basedHOTS / competency
A cuboid measures 5 cm by 3 cm by 2 cm. Find (i) its faces, vertices and edges and verify Euler's relation, (ii) the total length of all its edges, (iii) the area of its net.
Hint
Total edge length = 4(l + b + h).
Answer
(i) $6 + 8 - 12 = 2$ . (ii) $4 (5 + 3 + 2) = 40$ cm. (iii) $2 (15 + 6 + 10) = 62$ cm $^{2}$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Euler's relation?Show answer
F + V - E = 2.
2.Faces of a triangular prism?Show answer
5.
3.Edges of a cuboid?Show answer
12.
4.Vertices of a pyramid with n-sided base?Show answer
n + 1.

Euler's Relation and Nets of Solids: frequently asked questions

Does Euler's relation hold for a cylinder?

The relation is stated for solids with flat faces only.

What is a net?

A flat pattern that folds to make a solid.