NCERT Solutions for Class 6 Maths Chapter 5

Q.1 What is the disadvantage in comparing line segments by mere observation?

Ans. There are more chances of error due to improper viewing.

Q.2 Why is it better to use a divider than a ruler, while measuring the length of a line segment?

Ans. It is better to use a divider than a ruler because of accuracy in measuring the length of a line segment.

Q.3 Draw any line segment, say

$\overline{\mathrm{AB}}$

. Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB = AC + CB? [Note : If A,B,C are any three points on a line such that AC + CB = AB, then we can be sure that C lies between A and B.]

Ans.

Here, AB = 7 cm, AC = 5 cm and BC = 2 cm.

Yes, AB = AC + CB. (Since, 7 cm = 5 cm + 2 cm).

Q.4 If A, B, C are three points on a line such that AB = 5 cm, BC = 3 cm and AC = 8 cm, which one of them lies between the other two?

Ans.

Since, AB + BC = 5 cm + 3 cm
= 8 cm
= AC

So, B lies between A and C.

Q.5 

Verify,whetherDisthemidpointof\overline{AG}.

Ans.

Since AD = 3 and DG = 3.
So, D is mid-point of AG.

Q.6 If B is the mid point of

$\overline{\mathrm{AC}}$

and C is the mid point of

$\overline{\mathrm{BD}}$

, where A,B,C,D lie on a straight line, say why AB = CD?

Ans.

Since B is the mid-point of AC, so

AB = BC … (i)

And C is mid-point of BD, so

BC = CD … (ii)

Since BC is equal to AB and CD both.

So, AB = CD.

Q.7 Draw five triangles and measure their sides. Check in each case, if the sum of the lengths of any two sides is always less than the third side.

Ans.

In figure (a),

Sum of two sides = 5 cm + 5 cm

= 10 cm

Third side of triangle = 8 cm

So, sum of two sides is greater than third side of triangle.

In figure (b), Sum of two sides = 15 cm + 8 cm

= 23 cm

Third side of triangle = 17 cm

So, sum of two sides is greater than third side of triangle.

In figure (c),

Sum of two sides = 7 cm + 7 cm

= 14 cm

Third side of triangle = 11 cm

So, sum of two sides is greater than third side of triangle.
In figure (d),

Sum of two sides = 5.2 cm + 5.2 cm

= 10.4 cm

Third side of triangle = 5.2 cm

So, sum of two sides is greater than third side of triangle.

In figure (e),

Sum of two sides = 10 cm + 9 cm

= 19 cm

Third side of triangle = 15 cm

So, sum of two sides is greater than third side of triangle.

Therefore, the sum of the lengths of any two sides is always greater than the third side in a triangle.

Q.8 What fraction of a clockwise revolution does the hour hand of a clock turn through, when it goes from
(a) 3 to 9 (b) 4 to 7 (c) 7 to 10
(d) 12 to 9 (e) 1 to 10 (f) 6 to 3

Ans.

(a) From 3 to 9, the hour hand of a clock has made

$\frac{1}{2}$

of a revolution or 2 right angles.

(b) From 4 to 7, the hour hand of a clock has made

$\frac{1}{4}$

of a revolution or 1 right angle.

(c) From 7 to 10, the hour hand of a clock has made

$\frac{1}{4}$

of a revolution or 1 right angle.

(d) From 12 to 9, the hour hand of a clock has made

$\frac{3}{4}$

of a revolution or 3 right angles.

(e) From 1 to 10, the hour hand of a clock has made

$\frac{3}{4}$

of a revolution or 3 right angles.

(f) From 6 to 3, the hour hand of a clock has made

$\frac{3}{4}$

of a revolution or 3 right angles.

Q.9 Where will the hand of a clock stop if it
(a) starts at 12 and makes

$\frac{1}{2}$

of a revolution, clockwise?
(b) starts at 2 and makes

$\frac{1}{2}$

of a revolution, clockwise?
(c) starts at 5 and makes

$\frac{1}{4}$

of a revolution, clockwise?
(d) starts at 5 and makes

$\frac{3}{4}$

of a revolution, clockwise?

Ans.

$\begin{array}{l}\text{(a)}\\ \text{If the hand of a clock starts at 12 and makes}\\ \frac{1}{2}\text{of a revolution, clockwise, it will stop after}\\ \frac{1}{2}\times 12=6\text{hours}\\ \mathrm{i}.\mathrm{e}.\text{the hand of clock will stop at 6.}\\ \text{(b)}\\ \text{If the hand of a clock starts at 2 and makes}\\ \frac{1}{2}\text{of a revolution, clockwise, it will stop after}\\ \frac{1}{2}\times 12=6\text{hours}\\ \mathrm{i}.\mathrm{e}.\text{the hand of clock will stop at 8.}\\ \left(\mathrm{c}\right)\\ \text{If the hand of a clock starts at 5 and makes}\\ \frac{1}{4}\text{of a revolution, clockwise, it will stop after}\\ \frac{1}{4}\times 12=3\text{hours}\\ \mathrm{i}.\mathrm{e}.\text{the hand of clock will stop at 8.}\\ \left(\mathrm{d}\right)\\ \text{If the hand of a clock starts at 5 and makes}\\ \frac{3}{4}\text{of a revolution, clockwise, it will stop after}\\ \frac{3}{4}\times 12=9\text{hours}\\ \mathrm{i}.\mathrm{e}.\text{the hand of clock will stop at 2.}\end{array}$

Q.10 Which direction will you face if you start facing
(a) east and make

$\frac{1}{2}$

of a revolution clockwise?
(b) east and make

$1\frac{1}{2}$

of a revolution clockwise?
(c) west and make

$\frac{3}{4}$

of a revolution anti-clockwise?
(d) south and make one full revolution?
(Should we specify clockwise or anti-clockwise for this last question? Why not?)

Ans.

(a) If we are facing in east direction and make

$\frac{1}{2}$

of a revolution clockwise, we will face in west direction as shown :

(b) If we are facing in east direction and make

$1\frac{1}{2}$

of a revolution clockwise, we will face in west direction as shown :

(c) If we are facing in west direction and make

$\frac{3}{4}$

of a revolution anti-clockwise, we will face in north direction as shown :

(d) If we are facing in south direction and make one full revolution, we will face in south direction as shown:

There is no need to specify clockwise or anti-clockwise because if we make one complete revolution, we will be at the same place only.

Q.11 What part of a revolution have you turned through if you stand facing
(a) east and turn clockwise to face north?
(b) south and turn clockwise to face east?
(c) west and turn clockwise to face east?

Ans.

(a) If we stand facing east and turn clockwise to face north, we have completed

$\frac{3}{4}$

of a revolution.

(b) If we stand facing south and turn clockwise to face east, we have completed

$\frac{3}{4}$

of a revolution.

(d) If we stand facing west and turn clockwise to face east, we have completed

$\frac{1}{2}$

of a revolution.

Q.12 Find the number of right angles turned through by the hour hand of a clock when it goes from
(a) 3 to 6 (b) 2 to 8 (c) 5 to 11
(d) 10 to 1 (e) 12 to 9 (f) 12 to 6

Ans.

(a) If the hour hand of a clock goes from 3 to 6, it has turned 1 right angle.

(b) If the hour hand of a clock goes from 2 to 8, It has turned 2 right angles.

(c) If the hour hand of a clock goes from 5 to 11, it has turned 2 right angles.

(d) If the hour hand of a clock goes from 10 to 1, it has turned 1 right angle.

(e) If the hour hand of a clock goes from 12 to 9, it has turned 3 right angles.

(f) If the hour hand of a clock goes from 12 to 6, it has turned 2 right angles.

Q.13 How many right angles do you make if you start facing
(a) south and turn clockwise to west?
(b) north and turn anti-clockwise to east?
(c) west and turn to west?
(d) south and turn to north?

Ans.

(a) If we start facing south and turn clockwise to west, we have turned 1 right angle.

(b) If we start facing north and turn anti-clockwise to east, we have turned 3 right angles.

(c) If we start facing west and turn to west, we have turned 4 right angles.

(d) If we start facing south and turn to north(whether clockwise or anti- clockwise), we have turned 2 right angles.

Q.14 Where will the hour hand of a clock stop if it starts
(a) from 6 and turns through 1 right angle?
(b) from 8 and turns through 2 right angles?
(c) from 10 and turns through 3 right angles?
(d) from 7 and turns through 2 straight angles?

Ans.

(a) If the hour hand of a clock starts from 6 and turns through 1 right angle, it stops at 9.

(b) If the hour hand of a clock starts from 8 and turns through 2 right angles, it stops at 2.

(c) If the hour hand of a clock starts at 10 and turns through 3 right angles, it stops at 7.

(d) If the hour hand of a clock starts from 7 and turns through 2 straight angles, it stops at 7.

Q.15 Match the following:
(i) Straight angle (a) Less than one-fourth of a revolution
(ii) Right angle (b) More than half a revolution
(iii) Acute angle (c) Half of a revolution
(iv) Obtuse angle (d) One-fourth of a revolution
(v) Reflex angle (e) Between

$\frac{1}{4}$

and

$\frac{\text{1}}{\text{2}}$

of a revolution
(f) One complete revolution

Ans.

Q.16 Classify each one of the following angles as right, straight, acute, obtuse or reflex:



(f)

Ans.

(a) Acute angle

(b) Obtuse angle

(c) Right angle

(d) Reflex angle

(e) Straight angle

(f) Acute angle

Q.17 What is the measure of:
(i) a right angle?
(ii) a straight angle?

Ans.

(i) The measure of a right angle is 90°
(ii) The measure of a straight angle is 180°

Q.18 Say True or False :
(a) The measure of an acute angle < 90°.
(b) The measure of an obtuse angle < 90°.
(c) The measure of a reflex angle > 180°.
(d) The measure of one complete revolution = 360°.
(e) If m∠A = 53° and m∠B = 35°, then m∠A > m∠B.

Ans.

(a) True; acute angle is less than 90°.
(b) False; obtuse angle is greater than 90° and less than 180°.
(c) True; the measure of a reflex angle > 180°.
(d) True; the complete revolution is of 360°.
(e) True; angle A is larger than angle B.

Q.19 Write down the measures of
(a) some acute angles. (b) some obtuse angles. (give at least two examples of each).

Ans.

(a) 23° , 67°

(b) 91° , 120°

Q.20 Measure the angles given below using the Protractor and write down the measure.

Ans.

(a) 45°

(b) 120°

(c) 90°

(d) 60°, 130°, 90°

Q.21 Which angle has a large measure? First estimate and then measure.

Measure of Angle A =
Measure of Angle B =

Ans.

Measure of Angle A = 40°
Measure of Angle B = 60°
So, Angle B has larger measure.

Q.22 From these two angles which has larger measure? Estimate and then confirm by measuring them.

Ans.

Angle in part (b) seems larger.

Measure of angle in (a) = 45°

Measure of angle in (b) = 60°

Q.23 Fill in the blanks with acute, obtuse, right or straight :
(a) An angle whose measure is less than that of a right angle is______.
(b) An angle whose measure is greater than that of a right angle is ______.
(c) An angle whose measure is the sum of the measures of two right angles is _____.
(d) When the sum of the measures of two angles is that of a right angle, then each one of them is ______.
(e) When the sum of the measures of two angles is that of a straight angle and if one of them is acute then the other should be _______.

Ans.

(a) Acute angle
(b) Obtuse angle
(c) Straight angle
(d) Acute angle
(e) Obtuse angle

Q.24 Find the measure of the angle shown in each figure. (First estimate with your eyes and then find the actual measure with a protractor).

Ans.

Measure of given angle is 40°

Measure of given angle is 120°

Measure of given angle is 60°

Measure of given angle is 130°

Q.25 Find the angle measure between the hands of the clock in each figure:

Ans.

(a) Right angle; 90°

(b) Acute angle; 30°

(c) Straight angle; 180°

Q.26 Investigate

In the given figure, the angle measures 30°.
Look at the same figure through a magnifying glass.
Does the angle becomes larger? Does the size of the angle change?

Ans.

No, there is no change in the measurement of the Angle, if looked through a magnifying glass.

Q.27 Measure and classify each angle:

Ans.

Q.28 Which of the following are models for perpendicular lines :
(a) The adjacent edges of a table top.
(b) The lines of a railway track.
(c) The line segments forming the letter ‘L’.
(d) The letter V.

Ans.

(a) The adjacent edges of a table top are perpendicular lines as they form a right angle.

(b) The lines of a railway track do not form a right angle. As they are parallel lines and forms a zero angle.

(c) The line segments of the letter ‘L’ are perpendicular lines as they form a right angle.

(d) The letter V do not form a right angle but forms an acute angle.

Q.29

$\begin{array}{l}\mathrm{Let}\overline{\mathrm{PQ}}\mathrm{be}\mathrm{the}\mathrm{perpendicular}\mathrm{to}\mathrm{the}\mathrm{line}\mathrm{segment}\overline{\mathrm{XY}}.\\ \mathrm{Let}\overline{\mathrm{PQ}}\mathrm{and}\overline{\mathrm{XY}}\mathrm{intersect}\mathrm{in}\mathrm{the}\mathrm{point}\mathrm{A}.\mathrm{What}\mathrm{is}\mathrm{the}\\ \mathrm{measure}\mathrm{of}\angle \mathrm{PAY}?\end{array}$

Ans. The measure of angle PAY is 90°.

Q.30 There are two set-squares in your box. What are the measures of the angles that are formed at their corners? Do they have any angle measure that is common?

Ans. One set square is of 30°- 60°- 90° and another set square is of 45°- 45°- 90°. The right angle i.e. 90° is common between them.

Q.31 One set square is of 30°- 60°- 90° and another set square is of 45°- 45°- 90°. The right angle i.e. 90° is common between them.

Ans.

(a) Yes CE = EG; because C and G both are at equal distance from E.

(b) Yes; because E lies exactly half way between C and G.

(c)

$\overline{\mathrm{BH}}\text{and}\overline{\text{DF}}$

(d) (i) True; AC > FG

(ii) True; CD = GH

(iii) True; BC < EH

Q.32 Name the types of following triangles:
(a) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.
(b) ΔABC with AB = 8.7 cm, AC = 7 cm and BC = 6 cm.
(c) ΔPQR such that PQ = QR = PR = 5 cm.
(d) ΔDEF with m∠D= 90°
(e) ΔXYZ with m∠Y= 90° and XY = YZ.
(f) ΔLMN with m∠L = 30°, m∠M = 70° and m∠N= 80°.

Ans.

(a) All the sides of the triangle are unequal, so it is a scalene triangle.

(b) All the sides of triangle ABC are unequal, so triangle ABC is a scalene triangle.

(c) All the sides of triangle PQR are equal, so it is an equilateral triangle.

(d) In triangle DEF, angle D is of 90°, so it is a right-angled triangle.

(e) In triangle XYZ, two sides XY and YZ are equal and angle Y is of 90°, so it is an isosceles right triangle.

(f) In triangle LMN, all the angles are acute, so it is an acute-angled triangle.

Q.33 Match the following :
Measures of Triangle Type of Triangle
(i) 3 sides of equal length (a) Scalene
(ii) 2 sides of equal length (b) Isosceles right angled
(iii) All sides are of different length (c) Obtuse angled
(iv) 3 acute angles (d) Right angled
(v) 1 right angle (e) Equilateral
(vi) 1 obtuse angle (f) Acute angled
(vii) 1 right angle with two sides of equal length (g) Isosceles

Ans.

Measures of Triangle Type of Triangle
(i) 3 sides of equal length (e) Equilateral triangle
(ii) 2 sides of equal length (g)Isosceles
(iii) All sides are of different length (a)Scalene
(iv) 3 acute angles (f)Acute angled
(v) 1 right angle (d)Right angled
(vi) 1 obtuse angle (c)Obtuse angled
(vii) 1 right angle with two sides of equal length (b)Isosceles right angled

Q.34 Name each of the following triangles in two different ways: (you may judge the nature of the angle by observation)

Ans.

(a) This triangle has two sides of equal length and all angles are acute. So, it is an isosceles acute triangle.

(b) This triangle has all sides unequal and one right angle. So, it is a scalene-right triangle

(c) This triangle has two equal sides and one obtuse angle. So, it is an obtuse-isosceles triangle.

(d) This triangle has two equal sides and one right angle. So, it is an isosceles-right triangle.

(e) This triangle has all sides equal. So, it is an equilateral acute triangle.

(f) This triangle has all sides unequal and one obtuse angle. So, it is a scalene-obtuse triangle.

Q.35 Try to construct triangles using match sticks. Some are shown here. Can you make a triangle with
(a) 3 matchsticks?
(b) 4 matchsticks?
(c) 5 matchsticks?
(d) 6 matchsticks?
(Remember you have to use all the available matchsticks in each case) Name the type of triangle in each case. If you cannot make a triangle, think of reasons for it.

Ans.

(a)

(b) It is not possible to make triangle using four match sticks because sum of any two sides of a triangle is always greater than the third side.

(c)

(d)

Q.36 Say True or False:
(a) Each angle of a rectangle is a right angle.
(b) The opposite sides of a rectangle are equal in length.
(c) The diagonals of a square are perpendicular to one another.
(d) All the sides of a rhombus are of equal length.
(e) All the sides of a parallelogram are of equal length.
(f) The opposite sides of a trapezium are parallel.

Ans.

(a) True

(b) True

(c) True

(d) True

(e) False

(f) False

Q.37 Give reasons for the following:
(a) A square can be thought of as a special rectangle.
(b) A rectangle can be thought of as a special parallelogram.
(c) A square can be thought of as a special rhombus.
(d) Squares, rectangles, parallelograms are all quadrilaterals.
(e) Square is also a parallelogram.

Ans.

(a) A square is a special rectangle because a rectangle with all sides equal is a square.
(b) A rectangle is a special parallelogram because a parallelogram with all angle of measure 90° becomes a rectangle.
(c) A square is a special rhombus because a rhombus with all angles of measure 90° becomes a square.
(d) Squares, rectangles and parallelograms are quadrilaterals because they all have four sides.
(e) Square is also a parallelogram because its opposite sides are equal and parallel.

Q.38 A figure is said to be regular if its sides are equal in length and angles are equal in measure. Can you identify the regular quadrilateral?

Ans. A square has all sides equal and all angles of equal measure. So, square is a regular quadrilateral.

Q.39 Examine whether the following are polygons. If any one among them is not, say why?

Ans.

(a) The given figure is not closed, so it is not a polygon.
(b) The given figure is closed and has six line segments, so it is a polygon.
(c) The given figure is closed but has no line segment, so it is not a polygon.
(d) The given figure is closed but is joined by a curve, so it is not a polygon.

Q.40 Name each polygon.

Make two more examples of each of these.

Ans.

1. The given figure has four sides, so it is a quadrilateral. Example : A table top, book
2. The given figure has three sides, so it is a triangle. Example : Pizza slice, Sandwich
3. The given figure has five sides, so it is a pentagon. Example : Road signs, Rangoli patterns
4. The given figure has eight sides, so it is an octagon. Example : Bolt, Some clock designs.

Q.41 Draw a rough sketch of a regular hexagon. Connecting any three of its vertices, draw a triangle. Identify the type of the triangle you have drawn.

Ans.

This is the rough sketch of a regular hexagon.

The triangle drawn on connecting any three vertices of the hexagon is an isosceles obtuse triangle.

Q.42 Draw a rough sketch of a regular octagon. (Use squared paper if you wish). Draw a rectangle by joining exactly four of the vertices of the octagon.

Ans.

Q.43 A diagonal is a line segment that joins any two vertices of the polygon and is not a side of the polygon. Draw a rough sketch of a pentagon and draw its diagonals.

Ans.

Q.44 Match the following:

(a) Cone (i)

(b) Sphere (ii)

(c) Cylinder (iii)

(d) Cuboid (iv)

(e) Pyramid (v)

Give two new examples of each shape.

Ans.

(a) Cone
(b) Sphere
(c) Cylinder
(d) Cuboid
(e) Pyramid

Q.45 What shape is
(a) Your instrument box?
(b) A brick?
(c) A match box?
(d) A road-roller?
(e) A sweet laddu?

Ans.

(a) Cuboid
(b) Cuboid
(c) Cuboid
(d) Cylinder
(e) Sphere

Q.46 Find the perimeter of each of the following figures:

Ans.

(a) Perimeter of figure = 4 cm + 2 cm + 1 cm
+ 5 cm
= 12 cm
(b) Perimeter of figure = 23 cm + 35 cm + 35 cm
+ 40 cm
= 133 cm
(c) Perimeter of figure = 15 cm + 15 cm + 15 cm
+ 15 cm
= 60 cm
(d) Perimeter of figure = 4 cm + 4 cm + 4 cm
+ 4 cm + 4 cm
= 20 cm

(e) Perimeter of figure = 2.5 cm + 0.5 cm + 4 cm
+ 1 cm + 4 cm + 0.5 cm
+ 2.5 cm
= 15 cm
(f) Perimeter of figure = 4 cm + 1 cm + 3 cm
+ 2 cm+ 3 cm+ 4 cm+ 1 cm + 3 cm
+ 2 cm + 3 cm + 4 cm + 1 cm + 3 cm
+ 2 cm + 3 cm + 4 cm + 1 cm + 3 cm
+ 2 cm + 3 cm
= 52 cm

Q.47 The lid of a rectangular box of sides 40 cm by 10 cm is sealed all round with tape. What is the length of the tape required?

Ans. Since, lid of rectangular box is like a rectangle. So,

Length of tape = 2(length + Breadth)
= 2(40 + 10)
= 100 cm
= 1 m