CHAPTER

6 Geometry

Learning objectives
6.1 Point 6.2 Line
6.3 Surface 6.4 Length of Boundary of Plane figures

6.1 POINT
A point is represented by a dot. It has no size. Olympiad Bite
6.2 LINE Horizontal lines are also known
If we join two points with a ruler and extend it in both directions, as sleeping lines.
a line is formed. Vertical lines are also known as
standing lines.
P Q
The above line is represented by PQ .
Types of Lines
Vertical Lines

Horizontal Lines

Line Slanting Lines

Curved Lines

Olympiad Bite
Line Segment
The part of a line is called line segment. It has two fixed points and a A line segment cannot be
definite length. extend in any direction.
P Q
The above line segment is represented by PQ .
Ray
A ray starts from a fixed point and can be extended in the other direction. It has one end point and
no definite length.
P Q

The above ray is represented by PQ , where P is the initial point.
Parallel lines
Lines are said to be parallel, if they do not meet.
P Q
R S
PQ is parallel to RS .

Intersecting Lines
If two lines cross each other and meet at a point, then they are called intersecting lines.
P
R

O
Q
S
PQ and RS intersect each other at point O. Also, O is called the point of intersection.

SELF TEST - 1
1. How many horizontal lines are there in the (A) 5 (B) 3
given figure? (C) 4 (D) 6
(A) 5 (B) 4 4. The given figure shows a ______.
(C) 8 (D) 6
X Y
2. In the given figure, PQ and RS represent (A) Line (B) Line segment
_______ lines. (C) Ray (D) None of these
(A) Parallel R
(B) Intersecting 5. Which of the following line is intersecting
(C) Both (A) and (B) P Q LM ?
S
(D) Can't be determined
(A) XY X
L U
3. How many slanting lines are there in the given
(B) TU
figure?
(C) Both (A) and (B)
(D) Neither (A) nor (B) T M Y

6.3 SURFACE
The part of a solid which we can see and touch is called its Surface. Surface
h Plane Surface : Surface of a solid which are flat.
h Curved Surface : Surface of a solid which are not flat. Plane Surface Curved Surface
For example :
Flat Surface

Curved Surface

Flat Surface
Faces, Edges and Vertices
Olympiad Bite
In a solid object, we have
Vertex Only cube and cuboid has all flat faces.
Edges can be curved or straight.

Face

Edge

Solid Shapes Number of Faces Number of Edges Number of Vertices Example

6 12 8

Cube

6 12 8

Cuboid

3 2 0

Cylinder

2 1 1

Cone

1 0 0

Sphere

6.4 LENGTH OF BOUNDARY OF PLANE FIGURES

The sum of length of boundary of the given shapes are as follows:
4 cm 7 cm
Square Rectangle
4 cm

4 cm

3 cm

5
3 cm

cm

Triangle
3c

X Y
Z
4 cm 7 cm 7 cm
Square X
Olympiad Bite
Sum of all four sides = 4 cm + 4 cm + 4 cm + 4 cm = 16 cm
All four sides of square are of
Rectangle Y equal length.
Sum of all four sides = 7 cm + 3 cm + 7 cm + 3 cm = 20 cm Opposite sides of rectangle are of
equal length.
Triangle Z
Sum of all three sides = 3 cm + 5 cm + 7 cm = 15 cm

SELF TEST - 2
1. Which of the following is the best example of (A) 27 cm (B) 29 cm
a sphere? (C) 25 cm (D) 32 cm
(A) Gas cylinder (B) Geometry box 4. How many squares are there in the given
(C) Ball (D) Cube figure?
2. The given figure has ______ faces.

(A) 5 (B) 6
(C) 8 (D) 7
(A) 6 (B) 5
5. If the sum of length of boundary of the given
(C) 4 (D) 3
figure is 36 cm, then find the length of PS.
3. Find the sum of length of boundary of the S 10 cm R
given figure.

8 cm
11
c m
m
8c

P 10 cm Q
(A) 5 cm (B) 10 cm
10 cm (C) 8 cm (D) 12 cm
 EXERCISE
1. Which of the following shapes can be drawn 5. There are _____ triangles in the given figure.
using the given figure?

(A) 6 (B) 8
(C) 5 (D) 7
(A) (B)
6. Which of the following shapes has exactly
6 faces and 12 edges?

(C) (D) (A) (B)

2. Find the number of sides of the given figure. (C) (D) None of these

7. Which of the following is the best example of

cuboid?
(A) 15 (B) 10
(A) (B)
(C) 14 (D) 18
3. Which of the following shapes is missing in the
given figure?
12
(C) (D)
11 1
10 2

9 3
8. Which solid can be used to draw the shape X?
8 4
7 5
6

(A) Circle (B) Square Shape X

(C) Triangle (D) Rectangle
4. There are _______ faces and _____ vertices (A) (B)
respectively in the given figure.

(C) (D)
9. The shaded face of the given figure is ______.

(A) 8, 6 (B) 6, 8
(C) 12, 8 (D) 8, 12
(A) Rectangle (B) Square 15. Sumit wants to draw a square with the help of
(C) Circle (D) Triangle given objects. Which of the following objects can
he use?
10. The given figure is made up of _______ cubes.

P Q R
(A) Only P (B) Only Q
(A) 18 (B) 14
(C) Both P and Q (D) Both P and R
(C) 16 (D) 20
16. Which of the following figures have 6 flat faces?
11. Number of straight lines needed to make the
shaded figure is ________.
(A) (B)

(C) (D) None of these

17. There are _____ cylinders in the given figure.

(A) 15 (B) 12
(C) 18 (D) 20
12. There are _______ slanting lines in the given
figure.

(A) 9 (B) 8
(C) 6 (D) 7
18. Find the total length of boundary of the given
rectangle.
(A) 6 (B) 10
(C) 8 (D) 12 5 cm

13. How many faces does the given figure have? 10 cm

(A) 25 cm (B) 30 cm
(C) 20 cm (D) 40 cm
19. How many pairs of parallel lines are there in
the given figure?
(A) 4 (B) 5
(C) 9 (D) 6
14. The given figure is made up of 2 squares. Find
the length of each side of the square P. (A) 3 (B) 2
(C) 0 (D) 1
P 20. Identify me.
Q 5 cm
h I have 2 flat faces.
13 cm h I have 1 curved face.
(A) 10 cm (B) 8 cm h I don’t have any vertices.
(C) 12 cm (D) 13 cm h I have 2 curved edges.
 (A) 6 (B) 7
(C) 12 (D) 8
(A) (B)
25. How many pairs of intersecting lines are shown
in the given figure?

(C) (D)

21. Which of the following lines is parallel to line (a) (b) (c)
PQ?
(A) 2 (B) 1
M O T (C) 3 (D) 4
R
S 26. The number of vertices in the given figure is
_________.
P
Q
U N L

(A) Line RS (B) Line MN

(C) Line TU (D) Line OL
(A) 12 (B) 24
22. How many curved lines are there in the given
(C) 18 (D) 22
figure?
27. Which of the following figures have 3 faces and
2 edges?

(A) (B)
(A) 1 (B) 2
(C) 5 (D) 4
(C) (D)
23. A mouse has walked around the given triangle
completely. Find the distance covered by the mouse.
28. Sneha made the following sign board on topic
9c ‘NO PARKING’. Find the sum of length of boundary
m
m

of the board.
5c

15 cm
13 cm
7 cm
7 cm

(A) 18 cm (B) 25 cm No Parking

(C) 27 cm (D) 22 cm
24. How many triangles are there in the given cloud? 15 cm
(A) 54 cm (B) 44 cm
(C) 34 cm (D) 38 cm
29. What could be one of the faces of a cone?
(A) (B)
(C) Both (A) and (B) (D) Neither (A) nor (B)
30. Find the number of line segments in the given (A) 1-(b), 2-(a), 3-(c), 4-(d)
figure. (B) 1-(a), 2-(b), 3-(d), 4-(c)
(C) 1-(b), 2-(a), 3-(d), 4-(c)
(D) 1-(b), 2-(c), 3-(a), 4-(d)
33. Consider the following solids (made up of cubes)
(A) 12 (B) 18
and select the correct option.
(C) 15 (D) 20

31. Consider the figure made up of rectangle M

and identical squares N and O. Now answer the (a) (b)
following questions. (A) Number of cubes in fig. (a) is 10.
(B) Number of cubes in fig. (b) is 12.
4 cm (C) Total number of cubes in both figures is 23.
M N O
(D) None of these
34. Fill in the blanks and select the correct option.
20 cm
P. There are ________ faces in a cube.
(a) Find the length of longer side of rectangle M.
Q. Number of vertex in a sphere is ________.
(b) Find the difference between the length of R. In a solid object, two surfaces meet to make
boundary of rectangle M and square N. an _______.
(a) (b) P Q R
(A) 18 cm 28 cm (A) 6 0 Edge
(B) 12 cm 16 cm (B) 8 1 Edge
(C) 12 cm 25 cm (C) 5 0 Vertex
(D) 10 cm 28 cm (D) 6 1 Edge
32. Match the shapes given in Column-P with their 35. Study the signs of traffic rules used in India
properties given in Column-Q. given below.

Column-P Column-Q

1. (a) All sides are equal. Which of the following statements is CORRECT?
P. Each sign is made with straight lines only.
Q. There are 3 signs which are made with straight
lines only and one sign which is made with
2. (b) 3 sides and 3 corners. curved line.
(A) Only P (B) Only Q
(C) Both P and Q (D) Neither P nor Q
SOF IMO 2019 QUESTIONS
3. (c) No side and no corner.
1. Number of straight lines needed to make the
shaded figure is _______.

4. (d) Only opposite sides

are equal.
(A) 11 (B) 12 4. Which of the following figures has maximum
(C) 14 (D) 10 (Level-1) number of sides?
2. Suman wants to draw a circle with the help of
the given objects. (A) (B)

P Q R (C) (D)
Which of the following objects can she use?
(A) Only P (B) Both P and Q (Level-2)
(C) Both P and R (D) Both Q and R
 (Level-1) 5. A field is in the shape of a square. The sum of
all the sides of the field is 48 m. Find the length of
3. How many minimum straight lines are required
to draw the given figure? each side of the field.
(A) 26 (A) 10 m
(B) 20 (B) 14 m
(C) 24 (C) 8 m
(D) 32 (Level-1) (D) 12 m(Level-2)
 HINTS & EXPLANATIONS
SELF TEST - 1 12 Triangle
11 1
1. (B) : Horizontal lines are 10 2
J I G F Circle
JI, GF, AB and DE 3. (D): 9 3

So, number of horizontal H 8 4 Square

K L
lines = 4 C
7
6
5

A B D E 4. (B) : There are 6 faces and 8 vertices in the

2. (B) given figure.
F
5. (D): Triangles formed are T1, T2, T3, T4, T5,
A D T6 and T7 i.e., 7 in number.
3. (D):
G E T1 T2
B C
Slanting lines are AB, AG, GF, FE, DE and CD i.e., T3
6 in number. T6 T5
4. (C) T4
T7
5. (B) : TU is intersecting LM .
6. (A) : (A) It has 6 faces and 12 edges.
SELF TEST - 2 (B) It has 3 faces and 2 edges.
1. (C) (C) It has 2 faces and 1 edge.
2. (A) : A cuboid has 6 faces. 7. (B)  8. (B)
3. (B) : Sum of all sides = 8 cm + 10 cm + 11 cm 9. (B) : The shaded face of the given figure is
square.
= 29 cm
10. (C) :
4. (D): A B Layer 4
C D Layer 3
L Layer 2
M E F
K Layer 1
Number of cubes in the given figure
J I H G
= 6 + 6 + 2 + 2 = 16
Squares formed are : ABCL, LCMK, KMIJ, CDEM, C D
MEHI, EFGH and LDHJ. 11. (B) :
A B
So, number of squares formed = 7 L K
E F
5. (C) : Sum of all sides = PQ + QR + RS + SP
J I
⇒ 36 cm = 10 cm + 8 cm + 10 cm + SP
⇒ 36 cm = 28 cm + SP H G
⇒ SP = 36 cm – 28 cm = 8 cm Straight lines needed to make the shaded figure are
AB, BC, CD, DE, EF, FG, GH, HI, IJ, JK, KL and AL.
EXERCISE
So, 12 straight lines are needed to make the shaded
1. (B) figure.
2. (C) : Number of sides of the given figure = 14 12. (C)  13. (B)
14. (B) : As, Q is a square, so each side of square 30. (B) : Number of line segments formed in the
Q is 5 cm long. given figure = 18
G F 12 cm 4 cm 4 cm
E D

4 cm
31. (B) :

4 cm
P M N O
Q 5 cm
A B C 20 cm
13 cm
(a) Length of longer side of M + side of N + side of O
Also, AC = AB + BC
= 20 cm
⇒ 13 = AB + 5
⇒ AB = 8 cm Length of longer side of M + 4 + 4 = 20
So, each side of square P is 8 cm long. \ Length of longer side of M = 20 – 8 = 12 cm
(b) Length of boundary of rectangle M = 12 + 12 + 4 + 4
15. (A)
= 32 cm
16. (C) : It has 1 flat face. Length of boundary of square N = 4 + 4 + 4 + 4
(B) It has 1 flat face. = 16 cm
(C) It has 6 flat faces.
\ Required difference = 32 – 16 = 16 cm
17. (B)
32. (C) :
18. (B) : A rectangle has opposite sides of equal 1. Triangle : It has 3 sides and 3 corners.
length. 2. Square : Its all sides are equal.
Length of boundary of the given figure 3. Rectangle : Its opposite sides are equal.
= 5 cm +10 cm + 5 cm + 10 cm = 30 cm 4. Circle : It has no side and no corner.
A G
E 33. (C) : Number of cubes in fig.(a) = 6 + 6 = 12
19. (B) : F Number of cubes in fig.(b) = 6 + 4 + 1 = 11
\ Total number of cubes in both figures
D  = 12 + 11 = 23
B C
Pair of parallel lines are (AB, ED) and (AG, BC) 34. (A)  35. (B)
20. (B) : Curved edge Flat face
SOF IMO 2019 QUESTIONS
Curved face
1. (B) : Number of straight lines required to make
Curved edge Flat face the shaded figure = 12
2. (D)
21. (A) : Line RS is parallel to line PQ.
22. (B) 3. (C) : There are 8 horizontal, 8 vertical and 8
23. (C) : Distance covered by mouse slanting lines in the given figure.
 = (5 + 9 + 13 ) cm = 27 cm 4. (B) :
24. (D): There are 8 triangles in the given cloud. (A) Total number of sides = 12
(B) Total number of sides = 13
25. (C)  26. (B)
(C) Total number of sides = 10
27. (D): (A) It has 6 faces and 12 edges.
(D) Total number of sides = 12
(B) It has 2 faces and 1 edge.
So, figure in option (B) has maximum number of
(C) It has 1 face and 0 edge.
sides.
(D) It has 3 faces and 2 edges.
5. (D): A square has all four sides of equal length.
28. (B) : Sum of length of all sides of board = 7 cm
Sum of all four sides of square = 48 m
+ 15 cm + 7 cm + 15 cm = 44 cm
\ Each side of square = 48 ÷ 4
29. (B) = 12 m