MathsICSE Class 8 · Chapter 16

Representing 3-D in 2-D ICSE Class 8: Nets, Views and Euler's Relation

Updated 15 October 2026 · Fact-checked

Representing 3-D in 2-D means drawing solid shapes on flat paper as nets, views and sketches. ICSE Class 8 counts the faces (F), vertices (V) and edges (E) of cubes, cuboids, tetrahedrons, prisms and pyramids and verifies Euler's relation F + V − E = 2 for these flat-faced solids.

What Representing 3-D in 2-D is about

A solid (3-D shape) has length, breadth and height. Paper is flat (2-D), so we represent solids in several ways:

a net: the flat shape that folds up into the solid;
views: what you see from the front, the side and the top;
a sketch (oblique or isometric) that gives a sense of depth.

The CISCE Class VIII curriculum asks you to:

identify and match pictures with objects, including nested or joined shapes (not more than two),
draw 2-D representations of 3-D objects,
count vertices, edges and faces and verify Euler's relation for solids with flat faces: cubes, cuboids, tetrahedrons, prisms and pyramids.

A solid with only flat faces is a polyhedron. Its flat surfaces are faces, the line segments where two faces meet are edges, and the corners where edges meet are vertices. For every such solid in this chapter, F + V − E = 2.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Representing 3-D in 2-D sits in the Geometry theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Faces, edges and vertices

Face: a flat surface of the solid.
Edge: a line segment where two faces meet.
Vertex: a corner point where three or more edges meet.

Count carefully by working systematically: count the faces on the top, bottom and sides; count the vertices on the top and on the bottom; count edges top, bottom and up the sides.

Curved solids (cylinder, cone, sphere) are not polyhedrons, because they have curved surfaces.

Worked example

Count the faces, vertices and edges of a cube.

Try it first, then show the solutionSolution

Faces: top, bottom and 4 sides $= 6$ .
Vertices: 4 on top and 4 at the bottom $= 8$ .
Edges: 4 round the top, 4 round the bottom and 4 upright $= 12$ .

Answer: $F = 6$ , $V = 8$ , $E = 12$

Euler's relation

For every polyhedron in this chapter (cubes, cuboids, tetrahedrons, prisms, pyramids):
$F + V - E = 2$ .

Uses:

verify the relation from your counts (a good check that you counted correctly);
find a missing number: $E = F + V - 2$ , $V = E - F + 2$ , $F = E - V + 2$ ;
test whether a polyhedron with given numbers can exist: if $F + V - E \neq = 2$ , it cannot be one of these solids.

Worked example

Verify Euler's relation for a cube.

Try it first, then show the solutionSolution

$F = 6$ , $V = 8$ , $E = 12$ .
$F + V - E = 6 + 8 - 12 = 2$ ✓.

Answer: Verified: $6 + 8 - 12 = 2$ .

Prisms

A prism has two identical, parallel polygon ends (bases) joined by rectangular side faces. It is named after its base: triangular prism, pentagonal prism, and so on. A cuboid is a rectangular prism; a cube is a square prism with all edges equal.

For a prism whose base has $n$ sides:

faces $F = n + 2$ (two bases and $n$ rectangles),
vertices $V = 2 n$ ,
edges $E = 3 n$ .

Check: $(n + 2) + 2 n - 3 n = 2$ ✓ for every $n$ .

Worked example

Find $F$ , $V$ and $E$ for a triangular prism and a pentagonal prism.

Try it first, then show the solutionSolution

Triangular ( $n = 3$ ): $F = 5$ , $V = 6$ , $E = 9$ ; $5 + 6 - 9 = 2$ ✓.
Pentagonal ( $n = 5$ ): $F = 7$ , $V = 10$ , $E = 15$ ; $7 + 10 - 15 = 2$ ✓.

Answer: Triangular 5, 6, 9; pentagonal 7, 10, 15

Pyramids

A pyramid has one polygon base and triangular faces that meet at a single point, the apex. A tetrahedron is a triangular pyramid: all four faces are triangles.

For a pyramid whose base has $n$ sides:

faces $F = n + 1$ (base and $n$ triangles),
vertices $V = n + 1$ (base corners and the apex),
edges $E = 2 n$ (round the base and up to the apex).

Check: $(n + 1) + (n + 1) - 2 n = 2$ ✓.

Worked example

Find $F$ , $V$ and $E$ for a square pyramid and a tetrahedron, and verify Euler's relation.

Try it first, then show the solutionSolution

Square pyramid ( $n = 4$ ): $F = 5$ , $V = 5$ , $E = 8$ ; $5 + 5 - 8 = 2$ ✓.
Tetrahedron ( $n = 3$ ): $F = 4$ , $V = 4$ , $E = 6$ ; $4 + 4 - 6 = 2$ ✓.

Answer: Square pyramid 5, 5, 8; tetrahedron 4, 4, 6

Nets of solids

A net is a flat pattern that folds into a solid without overlaps or gaps.

Cube: 6 equal squares. There are 11 different nets; a common one is a cross shape (4 squares in a row with one square above and one below).
Cuboid: 6 rectangles in 3 matching pairs.
Triangular prism: 2 triangles and 3 rectangles.
Square pyramid: 1 square and 4 triangles, one on each side of the square.
Tetrahedron: 4 triangles (for example a large triangle divided into 4).

To test a net, imagine folding it: two faces must never land on the same place. Six squares in a straight line, for instance, cannot fold into a cube.

Worked example

Which solid has a net made of 2 triangles and 3 rectangles? Check with Euler's relation.

Try it first, then show the solutionSolution

Two identical triangular ends joined by 3 rectangles: a triangular prism.
$F = 5$ , $V = 6$ , $E = 9$ : $5 + 6 - 9 = 2$ ✓.

Answer: A triangular prism

Views and joined or nested shapes

Looking at a solid from three directions gives three 2-D views:

front view (from in front),
side view (from the left or right),
top view or plan (from directly above).

For a matchbox lying flat (5 cm × 3 cm × 1.5 cm): front view is a $5 \times 1.5$ rectangle, side view a $3 \times 1.5$ rectangle, top view a $5 \times 3$ rectangle.

Joined shapes are two solids stuck together (a cube with a pyramid on top, like a house). Nested shapes are one inside another (a ball in a cubical box). For a joined polyhedron, count the new solid: the shared face disappears, and shared edges and vertices are counted once.

Draw-and-label checklist: label each view (front, side, top), draw edges you can see as solid lines, keep the measurements to scale.

Worked example

Two identical square pyramids are joined base to base, so that the square bases match exactly. Count $F$ , $V$ and $E$ of the joined solid and verify Euler's relation.

Try it first, then show the solutionSolution

Faces: the two square bases are hidden, leaving $4 + 4 = 8$ triangles.
Vertices: the 4 corners of the shared square + 2 apexes $= 6$ .
Edges: 4 round the shared square (counted once) + 4 slant edges on each pyramid $= 12$ .
$8 + 6 - 12 = 2$ ✓.

Answer: $F = 8$ , $V = 6$ , $E = 12$

Maths in real life: Representing 3-D in 2-D

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.

Counting edges

Suppose a camping tent is shaped like a triangular prism with a floor, and a metal pole runs along every edge.

How many faces does the tent have, and how many poles are needed?

Try it first, then show the solutionSolution

A triangular prism has $F = 5$ (2 triangles, 3 rectangles).
Poles $=$ edges $= 9$ ; check $5 + 6 - 9 = 2$ ✓.

Answer: 5 faces; 9 poles

Memory hook: Edges are where faces meet: put a pole on each.

Net of a cuboid

Imagine you unfold a cardboard box measuring 20 cm × 15 cm × 10 cm into its net.

What shapes are in the net, and how much cardboard is used (ignoring flaps)?

Try it first, then show the solutionSolution

6 rectangles in pairs: $20 \times 15$ , $15 \times 10$ , $20 \times 10$ .
Area $= 2 (300 + 150 + 200) = 1300$ cm².

Answer: 3 pairs of rectangles; 1300 cm²

Memory hook: A cuboid's net is three matching pairs.

Joined solids

Suppose a toy house is a cube with a square pyramid roof sitting exactly on top.

How many edges does the toy house have?

Try it first, then show the solutionSolution

Cube: 12 edges; roof adds 4 slant edges (its base edges are the cube's top edges).
Total $= 16$ edges.

Answer: 16 edges

Memory hook: Count shared edges only once.

Topics in this chapter

Euler's Relation and Nets of Solids

Representing 3-D in 2-D formula sheet

Euler's relation: $F + V - E = 2$; Cubes, cuboids, prisms, pyramids, tetrahedrons.
Prism (n-sided base): $F = n + 2; V = 2 n; E = 3 n$; Cuboid: n = 4.
Pyramid (n-sided base): $F = n + 1; V = n + 1; E = 2 n$; Tetrahedron: n = 3.
Missing count: $E = F + V - 2$; Rearrange Euler's relation.

How the school exam asks Representing 3-D in 2-D

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Count and verify	Count F, V and E for a given solid and verify Euler's relation.	Count top, bottom and sides systematically; then add and subtract.
Find a missing number	A polyhedron has 12 faces and 30 edges. How many vertices?	Rearrange F + V − E = 2.
Can it exist?	Is a polyhedron with F = 10, E = 20, V = 15 possible?	Check whether F + V − E equals 2.
Nets	Identify the solid from its net, or draw a net.	Count the shapes in the net and imagine folding it.
Views and joined shapes	Draw the front, side and top views; count F, V, E of a joined solid.	Look from each direction; remove hidden shared faces when counting.

Tricks to remember Representing 3-D in 2-D

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

Faces + Vertices − Edges = 2
'Fun Visits End at 2': F + V − E = 2.
Use it when: Euler's relation.
Prism: 2, 2, 3
For a prism, F = n + 2, V = 2n, E = 3n.
Use it when: Prism counts.
Pyramid: 1, 1, 2
For a pyramid, F = n + 1, V = n + 1, E = 2n.
Use it when: Pyramid counts.
Glued faces vanish
When two solids are joined face to face, the shared faces disappear from the count.
Use it when: Joined shapes.

Representing 3-D in 2-D: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Edges of a prism · 1 mark · MCQExam level

The number of edges of a triangular prism is:
(a) 6
(b) 8
(c) 9
(d) 12

Try it first, then show the solutionSolution

Prism with $n = 3$ : $E = 3 n$ .
$E = 9$ .

Answer: (c)

Example 2: Square pyramid · 3 marks · Short answerExam level

Count the faces, vertices and edges of a square pyramid and verify Euler's relation.

Try it first, then show the solutionSolution

Faces: 1 square + 4 triangles $= 5$ .
Vertices: 4 base corners + apex $= 5$ ; edges: 4 base + 4 slant $= 8$ .
$5 + 5 - 8 = 2$ ✓.

Answer: $F = 5$ , $V = 5$ , $E = 8$ ; verified

Example 3: Hexagonal prism · 3 marks · Short answerExam level

Find $F$ , $V$ and $E$ for a hexagonal prism and verify Euler's relation.

Try it first, then show the solutionSolution

$n = 6$ : $F = 8$ (2 hexagons + 6 rectangles).
$V = 12$ , $E = 18$ .
$8 + 12 - 18 = 2$ ✓.

Answer: $F = 8$ , $V = 12$ , $E = 18$

Example 4: Missing vertices · 3 marks · Short answerHarder exam level

A polyhedron has 12 faces and 30 edges. How many vertices does it have?

Try it first, then show the solutionSolution

$F + V - E = 2$ .
$12 + V - 30 = 2$ .
$V = 20$ .

Answer: 20 vertices

Example 5: Is it possible? · 4 marks · Long answerHarder exam level

Can a polyhedron have 10 faces, 20 edges and 15 vertices? Give a reason.

Try it first, then show the solutionSolution

For a polyhedron, $F + V - E$ must equal 2.
$10 + 15 - 20 = 5$ .
$5 \neq = 2$ .
So no such polyhedron exists.

Answer: No, because $F + V - E = 5$ , not 2.

Example 6: A joined solid · 4 marks · Long answerHarder exam level

A square pyramid is placed exactly on the top face of a cube. Count the faces, vertices and edges of the new solid and verify Euler's relation.

Try it first, then show the solutionSolution

Faces: the cube's top is hidden, leaving 5; add the pyramid's 4 triangles: $F = 9$ .
Vertices: cube 8 + apex 1: $V = 9$ .
Edges: cube 12 + 4 slant edges: $E = 16$ .
$9 + 9 - 16 = 2$ ✓.

Answer: $F = 9$ , $V = 9$ , $E = 16$

Example 7: From edges to the shape · 5 marks · Long answerHOTS / competency

A prism has 21 edges. (i) What shape is its base? (ii) Find its faces and vertices. (iii) Find $F$ , $V$ , $E$ for a pyramid on the same base, and verify Euler's relation for both solids.

Try it first, then show the solutionSolution

$3 n = 21$ , so $n = 7$ : the base is a heptagon (7 sides).
(ii) Prism: $F = 9$ , $V = 14$ .
Check: $9 + 14 - 21 = 2$ ✓.
(iii) Pyramid: $F = 8$ , $V = 8$ , $E = 14$ .
Check: $8 + 8 - 14 = 2$ ✓.

Answer: Heptagonal base; prism 9, 14, 21; pyramid 8, 8, 14

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

The gift-box shop

Suppose a gift shop sells boxes in three shapes: a cube of edge 10 cm, a triangular prism and a square pyramid whose base edges are 10 cm and slant edges 12 cm. A ribbon is glued along every edge of a box.

(i) · 3 marks
Copy and complete a table of $F$ , $V$ and $E$ for the three boxes.
Answer
Answer: Cube 6, 8, 12; triangular prism 5, 6, 9; square pyramid 5, 5, 8
Cube: prism with $n = 4$ . Triangular prism: $n = 3$ , so $5, 6, 9$ . Square pyramid: $n = 4$ , so $5, 5, 8$ .
(ii) · 3 marks
Verify Euler's relation for each box.
Answer
Answer: Each gives 2
$6 + 8 - 12 = 2$ ; $5 + 6 - 9 = 2$ ; $5 + 5 - 8 = 2$ .
(iii) · 4 marks
Find the length of ribbon needed for the cube and for the pyramid.
Answer
Answer: Cube 120 cm; pyramid 88 cm
Cube: 12 edges × 10 cm $= 120$ cm. Pyramid: 4 base edges × 10 cm + 4 slant edges × 12 cm $= 40 + 48 = 88$ cm.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Count the faces, vertices and edges of a triangular prism and verify Euler's relation. (3 marks)

Step	Marks
$F = 5$ (2 triangles, 3 rectangles)	1
$V = 6$ , $E = 9$	1
$5 + 6 - 9 = 2$ ✓	1

A polyhedron has 8 faces and 6 vertices. Find the number of edges, and name a solid that fits. (4 marks)

Step	Marks
$F + V - E = 2$	1
$8 + 6 - E = 2$	1
$E = 12$	1
Example: two square pyramids joined base to base (an octahedron)	1

For a hexagonal pyramid, find $F$ , $V$ and $E$ , verify Euler's relation, and compare the number of edges with a hexagonal prism. (5 marks)

Step	Marks
$n = 6$ : $F = n + 1 = 7$	1
$V = 7$	1
$E = 2 n = 12$	1
$7 + 7 - 12 = 2$ ✓	1
Hexagonal prism has $E = 18$ , which is 6 more	1

Where students lose marks in Representing 3-D in 2-D

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

Mistake: Counting hidden faces of a joined solid
Faces that are glued together are no longer on the outside.
Wrong working
Cube + pyramid: $F = 6 + 5 = 11$
Right working
$F = 5 + 4 = 9$
Do this: Subtract the two glued faces.
Usually costs: The counts and the verification.
Mistake: Mixing up prism and pyramid formulas
A prism has two bases; a pyramid has one base and an apex.
Wrong working
Square pyramid: $V = 8$
Right working
Square pyramid: $V = 5$
Do this: Draw a quick sketch and count.
Usually costs: 1 to 2 marks.
Mistake: Writing Euler's relation as F + E − V = 2
The edges are subtracted.
Do this: F + V − E = 2: faces plus vertices minus edges.
Usually costs: The verification.
Mistake: Using Euler's relation for a cylinder or cone
These have curved surfaces; the relation is for solids with flat faces.
Do this: Use Euler's relation only for polyhedrons.
Usually costs: 1 mark.
Mistake: Unlabelled views
Examiners need to know which view is which.
Do this: Write 'front', 'side' and 'top' under each drawing.
Usually costs: 1 mark per view.

Practice questions on Representing 3-D in 2-D

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 · MCQExam level
The number of faces of a tetrahedron is:
(a) 3
(b) 4
(c) 5
(d) 6
Hint
Triangular pyramid.
Answer
(b)
Q2 · 1 mark · Very short answerExam level
Write $F$ , $V$ and $E$ for a cuboid.
Hint
Same as a cube.
Answer
$F = 6$ , $V = 8$ , $E = 12$ .
Q3 · 2 marks · Short answerExam level
Name the solid whose net has 2 triangles and 3 rectangles, and the solid whose net has 1 square and 4 triangles.
Hint
Count bases.
Answer
Triangular prism; square pyramid.
Q4 · 3 marks · Short answerHarder exam level
Find $F$ , $V$ and $E$ for a pentagonal pyramid and verify Euler's relation.
Hint
$n = 5$ .
Answer
$F = 6$ , $V = 6$ , $E = 10$ ; $6 + 6 - 10 = 2$ ✓.
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): A cube and a cuboid have the same numbers of faces, vertices and edges.
Reason (R): Both are prisms with four-sided bases.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Use the prism formulas with n = 4.
Answer
(c)
Q6 · 2 marks · Short answerHarder exam level
A polyhedron has 6 vertices and 12 edges. How many faces does it have?
Hint
$F = E - V + 2$ .
Answer
$F = 12 - 6 + 2 = 8$ .
Q7 · 3 marks · Short answerHarder exam level
A matchbox 5 cm long, 3 cm wide and 1.5 cm high lies flat on a table. Describe its front, side and top views with measurements.
Hint
Each view is a rectangle.
Answer
Front: $5 \times 1.5$ cm rectangle. Side: $3 \times 1.5$ cm rectangle. Top: $5 \times 3$ cm rectangle.
Q8 · 2 marks · Short answerHarder exam level
A square pyramid stands on a table. Describe its top view and its front view.
Hint
Look down at the apex.
Answer
Top view: a square with both diagonals drawn, meeting at the centre (the apex). Front view: a triangle.
Q9 · 4 marks · Long answerHarder exam level
Two identical cubes are glued together face to face. Count $F$ , $V$ and $E$ of the new solid and verify Euler's relation.
Hint
The result is a cuboid.
Answer
The new solid is a cuboid: $F = 6$ , $V = 8$ , $E = 12$ ; $6 + 8 - 12 = 2$ ✓. (The glued faces vanish, and the glued edges and vertices merge.)
Q10 · 3 marks · Short answerHarder exam level
A prism has 10 vertices. Find its faces and edges, and name it.
Hint
$2 n = 10$ .
Answer
$n = 5$ : pentagonal prism with $F = 7$ and $E = 15$ .
Q11 · 3 marks · Short answerHOTS / competency
A pyramid has 10 edges. What is the shape of its base? Find its faces and vertices and verify Euler's relation.
Hint
$2 n = 10$ .
Answer
$n = 5$ : pentagonal base. $F = 6$ , $V = 6$ ; $6 + 6 - 10 = 2$ ✓.
Q12 · 4 marks · Long answerHOTS / competency
A tetrahedron is placed exactly on one triangular end of a triangular prism. Count $F$ , $V$ and $E$ for the joined solid and verify Euler's relation.
Hint
Two triangular faces become hidden.
Answer
Faces: prism $5 - 1 = 4$ + tetrahedron $4 - 1 = 3$ , so $F = 7$ . Vertices: $6 + 1 = 7$ . Edges: $9 + 3 = 12$ . $7 + 7 - 12 = 2$ ✓.

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.State Euler's relation.Show answer
F + V − E = 2.
2.How many edges does a cube have?Show answer
12.
3.How many faces does a tetrahedron have?Show answer
4.
4.How many vertices does a pentagonal prism have?Show answer
10.
5.How many edges does a square pyramid have?Show answer
8.
6.What is a net?Show answer
A flat shape that folds up into a solid.
7.What shapes make the net of a square pyramid?Show answer
One square and four triangles.
8.What is the top view of a cylinder standing upright?Show answer
A circle.

Best order to study it

Learn the words face, edge and vertex, and count them for a cube and cuboid.
Learn the prism and pyramid patterns for F, V and E.
Verify Euler's relation and use it to find a missing number.
Practise nets of cubes, cuboids, prisms and pyramids.
Finish with views and joined or nested solids.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you count F, V and E without missing any?
Do you remember F = n + 2, V = 2n, E = 3n for prisms?
Do you remember F = V = n + 1, E = 2n for pyramids?
Can you rearrange F + V − E = 2 to find any one number?
Did you remove the glued faces in a joined solid?
Did you label every view?

Representing 3-D in 2-D: frequently asked questions

Does Euler's relation work for every solid?

It works for the flat-faced solids in this chapter (cubes, cuboids, prisms, pyramids, tetrahedrons) and many others. It is not used for curved solids like cylinders, cones and spheres.

What is the difference between a prism and a pyramid?

A prism has two identical parallel bases joined by rectangles. A pyramid has one base, and triangles that meet at a point called the apex.

How many nets does a cube have?

There are 11 different nets that fold into a cube. Not every arrangement of six squares works.

What are nested shapes?

Shapes placed one inside another, such as a ball inside a cubical box. Joined shapes are stuck together, such as a cube with a pyramid on top.

Why use front, side and top views?

One flat drawing cannot show every side of a solid. Three views together give the full shape, which is how builders and designers record objects.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII