Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 8

3-D Shapes, Euler's Rule, Nets and Symmetry

What to remember

A solid has faces, edges and vertices. Euler's rule for any simple solid with flat faces is F + V − E = 2.
A net is a flat pattern that folds into a solid. A cube has 11 different nets, and the 6 squares of a net must fold without overlap.
Line symmetry means one half is the mirror image of the other. Rotational symmetry means the figure looks the same before a full turn. Its order is the number of times this happens in one full turn.

Basic terms of solid shapes

A plane shape (2-D) has length and breadth only. A solid shape (3-D) has length, breadth and height, so it takes up space. A face is a flat or curved surface of a solid. An edge is the line where two faces meet. A vertex (plural: vertices) is the corner point where edges meet.

Common solids and the objects they look like:

Cube: a dice, a sugar cube.
Cuboid: a brick, a matchbox, a book.
Cylinder: a tin, a pipe, a candle.
Cone: an ice-cream cone, a joker's cap.
Sphere: a ball, a marble.
Prism: a tent shape, a Toblerone box (triangular prism).
Pyramid: the roof of a temple tower, a pyramid of Egypt (square pyramid).

A polyhedron (plural: polyhedra) is a solid whose faces are all flat polygons. Cube, cuboid, prism and pyramid are polyhedra. Cylinder, cone and sphere are not, because they have curved surfaces.

Faces, edges and vertices of common solids

Solid	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Triangular prism	5	9	6
Square pyramid	5	8	5
Triangular pyramid (tetrahedron)	4	6	4
Pentagonal prism	7	15	10
Hexagonal prism	8	18	12
Cylinder	3 (2 flat, 1 curved)	2 (curved)	0
Cone	2 (1 flat, 1 curved)	1 (curved)	1
Sphere	1 (curved)	0	0

Prism with an n-sided base: faces = n + 2, edges = 3n, vertices = 2n. The two bases are identical and parallel. The side faces are rectangles for a right prism.

Pyramid with an n-sided base: faces = n + 1, edges = 2n, vertices = n + 1. The side faces are triangles that meet at one point called the apex.

Euler's rule

Swiss mathematician Leonhard Euler showed that for every polyhedron without holes:

F + V − E = 2, which is the same as F + V = E + 2.

Check with a cube: 6 + 8 − 12 = 2. Check with a square pyramid: 5 + 5 − 8 = 2. Check with a triangular prism: 5 + 6 − 9 = 2.

Use the rule to find a missing number. A solid has 8 faces and 12 vertices. Then E = F + V − 2 = 8 + 12 − 2 = 18. This is a hexagonal prism.

The rule does not apply directly to curved solids. A cylinder would give 3 + 0 − 2 = 1, not 2, so we use the rule only for polyhedra.

Regular solids (Platonic solids) have all faces the same regular polygon and the same number of faces at each vertex:

Solid	Faces	Vertices	Edges	Face shape
Tetrahedron	4	4	6	Triangle
Cube	6	8	12	Square
Octahedron	8	6	12	Triangle
Dodecahedron	12	20	30	Pentagon
Icosahedron	20	12	30	Triangle

Nets of solids

A net is a 2-D pattern that folds along its edges to make a 3-D solid. Cutting and opening a box along its edges gives its net.

Cube: 6 equal squares. There are 11 different nets. In a row of four squares, the first and third are opposite faces, and so are the second and fourth.
Cuboid: 6 rectangles, opposite ones equal in size.
Cylinder: 2 circles and 1 rectangle. The length of the rectangle equals the circumference of the circle, and its breadth equals the height.
Cone: 1 circle and 1 sector of a bigger circle. The radius of the sector is the slant height of the cone.
Square pyramid: 1 square and 4 triangles.
Triangular prism: 2 triangles and 3 rectangles.
Tetrahedron: 4 triangles.

A cross-shaped net of 6 squares (a row of four with one square above and one below) folds into a cube. A straight row of six squares cannot fold into a cube. Always check that no two squares would land on the same face.

Views of solids

A solid looks different from different sides. We draw the front view, side view and top view. A cube looks like a square from every side. A cylinder standing upright looks like a rectangle from the front and a circle from the top. A cone standing on its base looks like a triangle from the front and a circle from the top. A sphere looks like a circle from every side.

Symmetry

Line (reflection) symmetry: A line divides the figure into two parts that match exactly when folded. This line is the line of symmetry or axis.

Figure	Lines of symmetry	Order of rotational symmetry
Equilateral triangle	3	3
Isosceles triangle	1	1 (none)
Scalene triangle	0	1 (none)
Square	4	4
Rectangle (not square)	2	2
Rhombus (not square)	2	2
Parallelogram (not rhombus or rectangle)	0	2
Kite	1	1 (none)
Isosceles trapezium	1	1 (none)
Regular pentagon	5	5
Regular hexagon	6	6
Circle	Infinite	Infinite
Semicircle	1	1 (none)

A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

Rotational symmetry: The figure fits onto itself after turning less than 360° about a fixed point called the centre of rotation. Angle of rotation = 360° ÷ order. A square turns by 90°, an equilateral triangle by 120°, a rectangle by 180°.

Point symmetry: A figure with rotational symmetry of order 2 looks the same when turned by 180°. Parallelogram, rectangle, rhombus, square and the letters H, I, N, O, S, X, Z have this.

Letters of the English alphabet:

Vertical line only: A, M, T, U, V, W, Y.
Horizontal line only: B, C, D, E, K.
Both lines: H, I, O, X.
Rotational symmetry only (no line): N, S, Z.
No symmetry at all: F, G, J, L, P, Q, R.

Symmetry in 3-D: A solid has symmetry about a plane. A cuboid (with three unequal edges) has 3 planes of symmetry, and a cube has 9.

Classroom angle

Children in Classes III to V learn solids from real objects. Let them sort objects, count faces and trace faces on paper. In Classes VI to VIII, they cut nets from card paper and fold them to check Euler's rule themselves. Paper folding makes line symmetry easy to see, and a pin through the centre of a cut-out helps children see rotational symmetry. Always move from the concrete object to the picture and then to the rule.

Exam traps

A cylinder has 3 faces, but only 2 are flat. A cone has 2 faces and 1 vertex.
A sphere has 1 curved face, no edge and no vertex.
Square pyramid has 5 vertices, but triangular prism has 6. Both have 5 faces.
A cube and a cuboid both have 6 faces, 12 edges and 8 vertices.
Euler's rule works for polyhedra only; do not apply it to the cylinder or cone.
A rectangle has 2 lines of symmetry, not 4; a parallelogram has none.
A rhombus has 2 lines of symmetry, and these are its diagonals. A rectangle's lines join the mid-points of opposite sides.
Order of rotational symmetry of a figure with no rotational symmetry is 1, not 0.

One-liners

Euler's rule: F + V − E = 2.
Cube: 6 faces, 12 edges, 8 vertices.
Tetrahedron: 4 faces, 6 edges, 4 vertices.
Prism with n-sided base has n + 2 faces.
Pyramid with n-sided base has 2n edges.
Cube has 11 nets.
Net of a cylinder: 2 circles and 1 rectangle.
Net of a cone: 1 circle and 1 sector.
Equilateral triangle has 3 lines of symmetry.
Circle has infinitely many lines of symmetry.
Angle of rotation = 360° ÷ order.
Letters H, I, O, X have both line and rotational symmetry.

Practice questions

How many edges does a cube have?
1. 12
2. 6
3. 8
4. 10
Answer
A. 12
A cube has 12 edges: 4 on top, 4 on the bottom and 4 vertical.
How many vertices does a square pyramid have?
1. 4
2. 6
3. 5
4. 8
Answer
C. 5
4 base corners plus 1 apex gives 5 vertices.
Euler's rule for polyhedra is
1. F + V + E = 2
2. F − V + E = 2
3. F + E − V = 0
4. F + V − E = 2
Answer
D. F + V − E = 2
Euler's rule: faces plus vertices minus edges equals 2.
How many faces does a triangular prism have?
1. 3
2. 4
3. 6
4. 5
Answer
D. 5
Two triangular bases and three rectangular sides make 5 faces.
The number of vertices of a cone is
1. 0
2. 2
3. 1
4. 3
Answer
C. 1
A cone has a single vertex, the apex.
A sphere has how many edges?
1. 0
2. 1
3. 2
4. 3
Answer
A. 0
A sphere has one curved surface, no edge and no vertex.
Which solid has 2 flat faces and 1 curved face?
1. Cone
2. Cylinder
3. Cuboid
4. Sphere
Answer
B. Cylinder
A cylinder has two flat circular bases and one curved surface.
The net of a cylinder consists of
1. 2 circles and 2 rectangles
2. 2 circles and 1 rectangle
3. 1 circle and 1 sector
4. 1 circle and 1 rectangle
Answer
B. 2 circles and 1 rectangle
The two bases are circles and the curved surface opens into a rectangle.
The net of a cone consists of
1. 1 circle and 1 sector
2. 2 circles and 1 rectangle
3. 1 circle and 1 triangle
4. 1 square and 4 triangles
Answer
A. 1 circle and 1 sector
The base is a circle and the curved surface opens into a sector.
How many lines of symmetry does an equilateral triangle have?
1. 1
2. 2
3. 3
4. 6
Answer
C. 3
One line passes through each vertex and the mid-point of the opposite side.
How many lines of symmetry does a rectangle (not a square) have?
1. 1
2. 2
3. 4
4. 0
Answer
B. 2
A rectangle has 2 lines, joining mid-points of opposite sides.
A parallelogram (not a rectangle or rhombus) has how many lines of symmetry?
1. 1
2. 2
3. 4
4. 0
Answer
D. 0
It has no line of symmetry, but it has rotational symmetry of order 2.
How many different nets does a cube have?
1. 11
2. 8
3. 6
4. 12
Answer
A. 11
A cube has exactly 11 distinct nets.
Which letter has both line symmetry and rotational symmetry?
1. N
2. H
3. E
4. A
Answer
B. H
H has vertical and horizontal lines and looks the same after a half turn.
Which letter has rotational symmetry but no line of symmetry?
1. S
2. T
3. B
4. M
Answer
A. S
S matches itself after a 180° turn but has no mirror line.
The order of rotational symmetry of a square is
1. 2
2. 4
3. 3
4. 8
Answer
B. 4
A square fits onto itself 4 times in one full turn.
The angle of rotation for an equilateral triangle is
1. 90°
2. 60°
3. 120°
4. 180°
Answer
C. 120°
Angle = 360° ÷ 3 = 120°.
A solid with 6 faces and 8 vertices has how many edges by Euler's rule?
1. 16
2. 14
3. 10
4. 12
Answer
D. 12
E = F + V − 2 = 6 + 8 − 2 = 12.
A polyhedron has 8 faces and 12 vertices. How many edges does it have?
1. 16
2. 20
3. 22
4. 18
Answer
D. 18
E = 8 + 12 − 2 = 18. It is a hexagonal prism.
A solid has 20 edges and 12 faces. How many vertices does it have?
1. 14
2. 8
3. 10
4. 12
Answer
C. 10
V = E + 2 − F = 20 + 2 − 12 = 10.
A prism has a base with 7 sides. How many faces does it have?
1. 9
2. 14
3. 7
4. 8
Answer
A. 9
Faces = n + 2 = 7 + 2 = 9.
A pyramid has a base with 6 sides. How many edges does it have?
1. 18
2. 6
3. 12
4. 7
Answer
C. 12
Edges = 2n = 12.
A prism has 24 edges. How many sides does its base have?
1. 6
2. 8
3. 7
4. 12
Answer
B. 8
Edges = 3n, so n = 24 ÷ 3 = 8.
The angle of rotation for a regular hexagon to look the same is
1. 60°
2. 90°
3. 120°
4. 30°
Answer
A. 60°
Angle = 360° ÷ 6 = 60°.
A cuboid with three unequal edges has how many planes of symmetry?
1. 3
2. 2
3. 6
4. 9
Answer
A. 3
One plane parallel to each pair of opposite faces gives 3.
How many planes of symmetry does a cube have?
1. 3
2. 6
3. 9
4. 12
Answer
C. 9
3 planes parallel to faces and 6 diagonal planes give 9.
A tin of milk standing on a table looks like what shape when seen from the top?
1. Rectangle
2. Triangle
3. Square
4. Circle
Answer
D. Circle
The top view of an upright cylinder is its circular base.
An upright cone looks like what shape from the front?
1. Square
2. Rectangle
3. Circle
4. Triangle
Answer
D. Triangle
The front view of a cone is a triangle.
A regular octahedron has how many vertices?
1. 8
2. 6
3. 12
4. 10
Answer
B. 6
An octahedron has 8 faces, 12 edges and 6 vertices.
The radius of the sector in the net of a cone equals the cone's
1. slant height
2. base radius
3. height
4. base diameter
Answer
A. slant height
The sector radius is the slant length from apex to the rim of the base.
The length of the rectangle in the net of a cylinder equals the
1. radius of the base
2. diameter of the base
3. area of the base
4. circumference of the base
Answer
D. circumference of the base
The curved surface wraps once around the base, so length = 2πr.
Which method is best for teaching children the faces and edges of solids in Class III?
1. Learning the rule by heart
2. Handling real objects like boxes and tins
3. Copying definitions from the board
4. Solving only written problems
Answer
B. Handling real objects like boxes and tins
Young learners move from concrete objects to pictures and then to rules.
Which of these has rotational symmetry of order 2 but no line of symmetry?
1. Rhombus
2. Rectangle
3. Parallelogram
4. Isosceles trapezium
Answer
C. Parallelogram
Only a parallelogram that is not a rectangle or rhombus has order 2 and no mirror line.
Statements about Euler's rule: 1. It holds for a cube. 2. It holds for a cylinder with F = 3, V = 0, E = 2. Which is/are correct?
1. Neither 1 nor 2
2. 1 only
3. 2 only
4. Both 1 and 2
Answer
B. 1 only
Cube: 6 + 8 − 12 = 2. Cylinder gives 1, so the rule is for polyhedra only.
Statements: 1. A cone has two faces. 2. A cone has two vertices. Which is/are correct?
1. Both 1 and 2
2. Neither 1 nor 2
3. 1 only
4. 2 only
Answer
C. 1 only
A cone has 1 flat and 1 curved face and only one vertex.
Statements: 1. A square has 4 lines of symmetry. 2. A rhombus has 4 lines of symmetry. Which is/are correct?
1. 2 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 1 only
Answer
D. 1 only
A rhombus (not a square) has only 2 lines, its diagonals.
Statements: 1. A circle has infinite lines of symmetry. 2. A circle has rotational symmetry about its centre for every angle. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Every diameter is a line of symmetry and any turn about the centre fits the circle.
Statements: 1. A straight row of six squares folds into a cube. 2. A cross made of six squares can fold into a cube. Which is/are correct?
1. Neither 1 nor 2
2. 1 only
3. 2 only
4. Both 1 and 2
Answer
C. 2 only
A row of six squares would overlap when folded; the cross net folds into a cube.
Statements: 1. Every prism with an n-sided base has 3n edges. 2. Every pyramid with an n-sided base has 3n edges. Which is/are correct?
1. Both 1 and 2
2. Neither 1 nor 2
3. 1 only
4. 2 only
Answer
C. 1 only
A pyramid has 2n edges, not 3n.
Statements: 1. The letter T has a vertical line of symmetry. 2. The letter B has a vertical line of symmetry. Which is/are correct?
1. 2 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 1 only
Answer
D. 1 only
B has a horizontal line of symmetry only.
Match the solid with its number of faces: P. Cube Q. Triangular prism R. Tetrahedron S. Pentagonal prism 1. 4 2. 5 3. 6 4. 7
1. P-3, Q-2, R-1, S-4
2. P-2, Q-3, R-1, S-4
3. P-3, Q-2, R-4, S-1
4. P-3, Q-1, R-2, S-4
Answer
A. P-3, Q-2, R-1, S-4
Cube 6, triangular prism 5, tetrahedron 4, pentagonal prism 7.
Match the figure with its lines of symmetry: P. Square Q. Isosceles triangle R. Regular pentagon S. Rectangle 1. 1 2. 2 3. 4 4. 5
1. P-4, Q-1, R-3, S-2
2. P-3, Q-1, R-4, S-2
3. P-2, Q-1, R-4, S-3
4. P-3, Q-2, R-4, S-1
Answer
B. P-3, Q-1, R-4, S-2
Square 4, isosceles triangle 1, regular pentagon 5, rectangle 2.
Match the solid with its number of edges: P. Square pyramid Q. Cube R. Hexagonal prism S. Cone 1. 1 2. 8 3. 12 4. 18
1. P-2, Q-3, R-4, S-1
2. P-2, Q-4, R-3, S-1
3. P-3, Q-2, R-4, S-1
4. P-2, Q-3, R-1, S-4
Answer
A. P-2, Q-3, R-4, S-1
Square pyramid 8, cube 12, hexagonal prism 18, cone 1 curved edge.
Match the figure with the order of rotational symmetry: P. Rectangle Q. Equilateral triangle R. Regular hexagon S. Scalene triangle 1. 1 2. 2 3. 3 4. 6
1. P-2, Q-3, R-4, S-1
2. P-3, Q-2, R-4, S-1
3. P-2, Q-3, R-1, S-4
4. P-4, Q-3, R-2, S-1
Answer
A. P-2, Q-3, R-4, S-1
Rectangle 2, equilateral triangle 3, regular hexagon 6, scalene triangle 1.
Which solid has the same number of faces and vertices?
1. Cube
2. Square pyramid
3. Cuboid
4. Triangular prism
Answer
B. Square pyramid
The square pyramid has 5 faces and 5 vertices.

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1.A solid is in the shape of a cylinder of radius 7 units and height 12 units, surmounted by a cone of the same radius and height 9 units. Find the total volume of the solid. (Use pi = 22/7)
2.A solid is in the shape of a cylinder of radius 10.5 units and height 22 units, surmounted by a cone of the same radius and height 13 units. Find the total volume of the solid. (Use pi = 22/7)
3.A solid is in the shape of a cylinder of radius 7 units and height 12 units, surmounted by a cone of the same radius and height 11 units. Find the total volume of the solid. (Use pi = 22/7)
4.A solid is in the shape of a cylinder of radius 14 units and height 24 units, surmounted by a cone of the same radius and height 9 units. Find the total volume of the solid. (Use pi = 22/7)
5.A solid is in the shape of a cylinder of radius 10.5 units and height 10 units, surmounted by a cone of the same radius and height 13 units. Find the total volume of the solid. (Use pi = 22/7)
6.A solid is in the shape of a cylinder of radius 7 units and height 14 units, surmounted by a cone of the same radius and height 11 units. Find the total volume of the solid. (Use pi = 22/7)
7.A solid is in the shape of a cylinder of radius 7 units and height 16 units, surmounted by a cone of the same radius and height 9 units. Find the total volume of the solid. (Use pi = 22/7)
8.A solid is in the shape of a cylinder of radius 14 units and height 24 units, surmounted by a cone of the same radius and height 11 units. Find the total volume of the solid. (Use pi = 22/7)
9.A solid is in the shape of a cylinder of radius 14 units and height 22 units, surmounted by a cone of the same radius and height 9 units. Find the total volume of the solid. (Use pi = 22/7)
10.A solid is in the shape of a cylinder of radius 14 units and height 18 units, surmounted by a cone of the same radius and height 9 units. Find the total volume of the solid. (Use pi = 22/7)

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