Mathematics NCERT Grade 10, Chapter 12: Areas related to circles- We already know the methods of finding perimeter and area of simple plane figures such as rectanglessquares, parallelograms, triangles, and circles. In this chapter, we will study about the areas related to circles. The first section is about the perimeter and area of a circle.
• Area of the circle = 1/2 $×$ 2$\mathrm{\pi }$r $×$r $\mathrm{\pi }$r2
Great Indian mathematician Aryabhata gave an approximate value of  (pie) $\mathrm{\pi }$ which is a constant ratio that the circumference has with the diameter of the circle One solved example is discussed in this section in which the cost of ploughing the field is found and the student thus recalls the formulas for finding the circumference and area of a circle. First exercise 12.1 is a short exercise containing 5 questions consisting of word problems.
Second section deals with areas of sector and segment of a circle.
Two types of the sectors are discussed here:
• Major sector
• Minor sector
Two types of the segments are discussed here:
• Minor segment
• Major segment
Area of a segment of a circle can be calculated by subtracting the area of the corresponding triangle from the area of the corresponding sector.
Two examples are discussed in this section to make the concept clear to students. Exercise 12.2 contains 14 questions that are related to the area of circles
The next section is about the topic- Areas of combination of plane figures. In the following exercise, a student will study how to find the area of some combinations of plane figures like flower beds, window designs, table covers etc. This particular section contains 3 solved examples and then the last exercise of the chapter -12.3 is given. Most of the questions in this exercise are figure based.
The chapter is summarised at the end in the form of formulas and short notes.

#### Question 1:

The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Radius (r1) of 1st circle = 19 cm

Radius (r2) or 2nd circle = 9 cm

Let the radius of 3rd circle be r.

Circumference of 1st circle = 2πr1 = 2π (19) = 38π

Circumference of 2nd circle = 2πr2 = 2π (9) = 18π

Circumference of 3rd circle = 2πr

Given that,

Circumference of 3rd circle = Circumference of 1st circle + Circumference of 2nd circle

r = 38π + 18π = 56π

Therefore, the radius of the circle which has circumference equal to the sum of the circumference of the given two circles is 28 cm.

##### Video Solution for real numbers (Page: 225 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 225 , Question 1

#### Question 2:

The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Radius (r1) of 1st circle = 8 cm

Radius (r2) of 2nd circle = 6 cm

Let the radius of 3rd circle be r.

Area of 1st circle

Area of 2nd circle

Given that,

Area of 3rd circle = Area of 1st circle + Area of 2nd circle

However, the radius cannot be negative. Therefore, the radius of the circle having area equal to the sum of the areas of the two circles is 10 cm.

##### Video Solution for real numbers (Page: 225 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 225 , Question 2

#### Question 3:

Given figure depicts an archery target marked with its five scoring areas from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing Gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

Radius (r1) of gold region (i.e., 1st circle)

Given that each circle is 10.5 cm wider than the previous circle.

Therefore, radius (r2) of 2nd circle = 10.5 + 10.5

21 cm

Radius (r3) of 3rd circle = 21 + 10.5

= 31.5 cm

Radius (r4) of 4th circle = 31.5 + 10.5

= 42 cm

Radius (r5) of 5th circle = 42 + 10.5

= 52.5 cm

Area of gold region = Area of 1st circle

Area of red region = Area of 2nd circle − Area of 1st circle

Area of blue region = Area of 3rd circle − Area of 2nd circle

Area of black region = Area of 4th circle − Area of 3rd circle

Area of white region = Area of 5th circle − Area of 4th circle

Therefore, areas of gold, red, blue, black, and white regions are 346.5 cm2, 1039.5 cm2, 1732.5 cm2, 2425.5 cm2, and 3118.5 cm2 respectively.

##### Video Solution for real numbers (Page: 225 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 225 , Question 3

#### Question 4:

The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is traveling at a speed of 66 km per hour?

Diameter of the wheel of the car = 80 cm

Radius (r) of the wheel of the car = 40 cm

Circumference of wheel = 2πr

= 2π (40) = 80π cm

Speed of car = 66 km/hour

Distance travelled by the car in 10 minutes

= 110000 × 10 = 1100000 cm

Let the number of revolutions of the wheel of the car be n.

n × Distance travelled in 1 revolution (i.e., circumference)

= Distance travelled in 10 minutes

Therefore, each wheel of the car will make 4375 revolutions.

##### Video Solution for real numbers (Page: 226 , Q.No.: 4)

NCERT Solution for Class 10 math - real numbers 226 , Question 4

#### Question 5:

Tick the correct answer in the following and justify your choice: If the perimeter and the area of a circle are numerically equal, then the radius of the circle is

(A) 2 units (B) π units (C) 4 units (D)7 units

Let the radius of the circle be r.

Circumference of circle = 2πr

Area of circle = πr2

Given that, the circumference of the circle and the area of the circle are equal.

This implies 2πr = πr2

2 = r

Therefore, the radius of the circle is 2 units.

Hence, the correct answer is A.

##### Video Solution for real numbers (Page: 226 , Q.No.: 5)

NCERT Solution for Class 10 math - real numbers 226 , Question 5

#### Question 1:

Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.

Let OACB be a sector of the circle making 60° angle at centre O of the circle.

Area of sector of angle θ =

Area of sector OACB =

Therefore, the area of the sector of the circle making 60° at the centre of the circle is

##### Video Solution for real numbers (Page: 230 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 230 , Question 1

#### Question 2:

Find the area of a quadrant of a circle whose circumference is 22 cm.

Let the radius of the circle be r.

Circumference = 22 cm

r = 22

Quadrant of circle will subtend 90° angle at the centre of the circle.

Area of such quadrant of the circle

##### Video Solution for real numbers (Page: 230 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 230 , Question 2

#### Question 3:

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

We know that in 1 hour (i.e., 60 minutes), the minute hand rotates 360°.

In 5 minutes, minute hand will rotate =

Therefore, the area swept by the minute hand in 5 minutes will be the area of a sector of 30° in a circle of 14 cm radius.

Area of sector of angle θ =

Area of sector of 30°

Therefore, the area swept by the minute hand in 5 minutes is

##### Video Solution for real numbers (Page: 230 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 230 , Question 3

#### Question 4:

A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding:

(i) Minor segment

(ii) Major sector

[Use π = 3.14]

Let AB be the chord of the circle subtending 90° angle at centre O of the circle.

Area of major sector OADB =

Area of minor sector OACB =

Area of ΔOAB =

= 50 cm2

Area of minor segment ACB = Area of minor sector OACB −

Area of ΔOAB = 78.5 − 50 = 28.5 cm2

##### Video Solution for real numbers (Page: 230 , Q.No.: 4)

NCERT Solution for Class 10 math - real numbers 230 , Question 4

#### Question 5:

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:

(i) The length of the arc

(ii) Area of the sector formed by the arc

(iii) Area of the segment forced by the corresponding chord

Radius (r) of circle = 21 cm

Angle subtended by the given arc = 60°

Length of an arc of a sector of angle θ =

Length of arc ACB =

= 22 cm

Area of sector OACB =

In ΔOAB,

∠OAB = ∠OBA (As OA = OB)

∠OAB + ∠AOB + ∠OBA = 180°

2∠OAB + 60° = 180°

∠OAB = 60°

Therefore, ΔOAB is an equilateral triangle.

Area of ΔOAB =

Area of segment ACB = Area of sector OACB − Area of ΔOAB

##### Video Solution for real numbers (Page: 230 , Q.No.: 5)

NCERT Solution for Class 10 math - real numbers 230 , Question 5

#### Question 6:

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle.

[Use π = 3.14 and]

Radius (r) of circle = 15 cm

Area of sector OPRQ =

In ΔOPQ,

∠OPQ = ∠OQP (As OP = OQ)

∠OPQ + ∠OQP + ∠POQ = 180°

2 ∠OPQ = 120°

∠OPQ = 60°

ΔOPQ is an equilateral triangle.

Area of ΔOPQ =

Area of segment PRQ = Area of sector OPRQ − Area of ΔOPQ

= 117.75 − 97.3125

= 20.4375 cm2

Area of major segment PSQ = Area of circle − Area of segment PRQ

##### Video Solution for real numbers (Page: 230 , Q.No.: 6)

NCERT Solution for Class 10 math - real numbers 230 , Question 6

#### Question 7:

A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle.

[Use π = 3.14 and]

Let us draw a perpendicular OV on chord ST. It will bisect the chord ST.

SV = VT

In ΔOVS,

Area of ΔOST =

Area of sector OSUT =

Area of segment SUT = Area of sector OSUT − Area of ΔOST

= 150.72 − 62.28

= 88.44 cm2

##### Video Solution for real numbers (Page: 230 , Q.No.: 7)

NCERT Solution for Class 10 math - real numbers 230 , Question 7

#### Question 8:

A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see the given figure). Find

(i) The area of that part of the field in which the horse can graze.

(ii) The increase in the grazing area of the rope were 10 m long instead of 5 m.

[Use π = 3.14]

From the figure, it can be observed that the horse can graze a sector of 90° in a circle of 5 m radius.

Area that can be grazed by horse = Area of sector OACB

Area that can be grazed by the horse when length of rope is 10 m long

Increase in grazing area = (78.5 − 19.625) m2

= 58.875 m2

##### Video Solution for real numbers (Page: 230 , Q.No.: 8)

NCERT Solution for Class 10 math - real numbers 230 , Question 8

#### Question 9:

A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in figure. Find.

(i) The total length of the silver wire required.

(ii) The area of each sector of the brooch

Total length of wire required will be the length of 5 diameters and the circumference of the brooch.

Circumference of brooch = 2πr

= 110 mm

Length of wire required = 110 + 5 × 35

= 110 + 175 = 285 mm

It can be observed from the figure that each of 10 sectors of the circle is subtending 36° at the centre of the circle.

Therefore, area of each sector =

##### Video Solution for real numbers (Page: 230 , Q.No.: 9)

NCERT Solution for Class 10 math - real numbers 230 , Question 9

#### Question 10:

An umbrella has 8 ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

There are 8 ribs in an umbrella. The area between two consecutive ribs is subtending at the centre of the assumed flat circle.

Area between two consecutive ribs of circle =

##### Video Solution for real numbers (Page: 231 , Q.No.: 10)

NCERT Solution for Class 10 math - real numbers 231 , Question 10

#### Question 11:

A car has two wipers which do not overlap. Each wiper has blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades.

It can be observed from the figure that each blade of wiper will sweep an area of a sector of 115° in a circle of 25 cm radius.

Area of such sector =

Area swept by 2 blades =

##### Video Solution for real numbers (Page: 231 , Q.No.: 11)

NCERT Solution for Class 10 math - real numbers 231 , Question 11

#### Question 12:

To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships warned. [Use π = 3.14]

It can be observed from the figure that the lighthouse spreads light across a

sector of 80° in a circle of 16.5 km radius.

Area of sector OACB =

##### Video Solution for real numbers (Page: 231 , Q.No.: 12)

NCERT Solution for Class 10 math - real numbers 231 , Question 12

#### Question 13:

A round table cover has six equal designs as shown in figure. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs.0.35 per cm2. [Use]

It can be observed that these designs are segments of the circle.

Consider segment APB. Chord AB is a side of the hexagon. Each chord will substitute at the centre of the circle.

In ΔOAB,

∠OAB = ∠OBA (As OA = OB)

∠AOB = 60°

∠OAB + ∠OBA + ∠AOB = 180°

2∠OAB = 180° − 60° = 120°

∠OAB = 60°

Therefore, ΔOAB is an equilateral triangle.

Area of ΔOAB =

= 333.2 cm2

Area of sector OAPB =

Area of segment APB = Area of sector OAPB − Area of ΔOAB

Cost of making 1 cm2 designs = Rs 0.35

Cost of making 464.76 cm2 designs = = Rs 162.68

Therefore, the cost of making such designs is Rs 162.68.

##### Video Solution for real numbers (Page: 231 , Q.No.: 13)

NCERT Solution for Class 10 math - real numbers 231 , Question 13

#### Question 14:

Tick the correct answer in the following:

Area of a sector of angle p (in degrees) of a circle with radius R is

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

We know that area of sector of angle θ =

Area of sector of angle P =

Hence, (D) is the correct answer.

##### Video Solution for real numbers (Page: 231 , Q.No.: 14)

NCERT Solution for Class 10 math - real numbers 231 , Question 14

#### Question 1:

Find the area of the shaded region in the given figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.

It can be observed that RQ is the diameter of the circle. Therefore, ∠RPQ will be 90º.

By applying Pythagoras theorem in ΔPQR,

RP2 + PQ2 = RQ2

(7)2 + (24)2 = RQ2

Since RQ is the diameter of the circle, it divides the circle in two equal parts.

Area of ΔPQR

Area of shaded region = Area of semi-circle RPQOR − Area of ΔPQR

=

=

= cm2

##### Video Solution for real numbers (Page: 234 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 234 , Question 1

#### Question 2:

Find the area of the shaded region in the given figure, if radii of the two concentric circles with centre O are 7 cm and 14 cm respectively and ∠AOC = 40°.

Radius of inner circle = 7 cm

Radius of outer circle = 14 cm

Area of shaded region = Area of sector OAFC − Area of sector OBED

$=\frac{40°}{360°}×\mathrm{\pi }\left(14{\right)}^{2}-\frac{40°}{360°}×\mathrm{\pi }\left(7{\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{1}{9}×\frac{22}{7}×14×14-\frac{1}{9}×\frac{22}{7}×7×7\phantom{\rule{0ex}{0ex}}=\frac{616}{9}-\frac{154}{9}=\frac{462}{9}\phantom{\rule{0ex}{0ex}}=\frac{154}{3}{\mathrm{cm}}^{2}$

##### Video Solution for real numbers (Page: 235 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 235 , Question 2

#### Question 3:

Find the area of the shaded region in the given figure, if ABCD is a square of side 14 cm and APD and BPC are semicircles.

It can be observed from the figure that the radius of each semi-circle is 7 cm.

Area of each semi-circle =

Area of square ABCD = (Side)2 = (14)2 = 196 cm2

= Area of square ABCD − Area of semi-circle APD − Area of semi-circle BPC

= 196 − 77 − 77 = 196 − 154 = 42 cm2

##### Video Solution for real numbers (Page: 235 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 235 , Question 3

#### Question 4:

Find the area of the shaded region in the given figure, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.

We know that each interior angle of an equilateral triangle is of measure 60°.

Area of sector OCDE

Area of

Area of circle = πr2

Area of shaded region = Area of ΔOAB + Area of circle − Area of sector OCDE

##### Video Solution for real numbers (Page: 235 , Q.No.: 4)

NCERT Solution for Class 10 math - real numbers 235 , Question 4

#### Question 5:

From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in the given figure. Find the area of the remaining portion of the square.

Each quadrant is a sector of 90° in a circle of 1 cm radius.

Area of square = (Side)2 = (4)2 = 16 cm2

Area of circle = πr2 = π (1)2

Area of the shaded region = Area of square − Area of circle − 4 × Area of quadrant

##### Video Solution for real numbers (Page: 235 , Q.No.: 5)

NCERT Solution for Class 10 math - real numbers 235 , Question 5

#### Question 6:

In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in the given figure. Find the area of the design (Shaded region).

Radius (r) of circle = 32 cm

AD is the median of ABC.

In ΔABD,

Area of equilateral triangle,

Area of circle = πr2

Area of design = Area of circle − Area of ΔABC

##### Video Solution for real numbers (Page: 235 , Q.No.: 6)

NCERT Solution for Class 10 math - real numbers 235 , Question 6

#### Question 7:

In the given figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

Area of each of the 4 sectors is equal to each other and is a sector of 90° in a circle of 7 cm radius.

Area of each sector

Area of square ABCD = (Side)2 = (14)2 = 196 cm2

Area of shaded portion = Area of square ABCD − 4 × Area of each sector

Therefore, the area of shaded portion is 42 cm2.
##### Video Solution for real numbers (Page: 236 , Q.No.: 7)

NCERT Solution for Class 10 math - real numbers 236 , Question 7

#### Question 8:

Thegivenfigure depicts a racing track whose left and right ends are semicircular.

The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find:

(i) The distance around the track along its inner edge

(ii) The area of the track

Distance around the track along its inner edge = AB + arc BEC + CD + arc DFA

Area of the track = (Area of GHIJ − Area of ABCD) + (Area of semi-circle HKI − Area of semi-circle BEC) + (Area of semi-circle GLJ − Area of semi-circle AFD)

Therefore, the area of the track is 4320 m2.

##### Video Solution for real numbers (Page: 236 , Q.No.: 8)

NCERT Solution for Class 10 math - real numbers 236 , Question 8

#### Question 9:

In the given figure, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

Radius (r1) of larger circle = 7 cm

Area of smaller circle

Area of semi-circle AECFB of larger circle

Area of

= Area of smaller circle + Area of semi-circle AECFB − Area of ΔABC

##### Video Solution for real numbers (Page: 236 , Q.No.: 9)

NCERT Solution for Class 10 math - real numbers 236 , Question 9

#### Question 10:

The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (See the given figure). Find the area of shaded region. [Use π = 3.14 and]

Let the side of the equilateral triangle be a.

Area of equilateral triangle = 17320.5 cm2

Each sector is of measure 60°.

Area of shaded region = Area of equilateral triangle − 3 × Area of each sector

##### Video Solution for real numbers (Page: 236 , Q.No.: 10)

NCERT Solution for Class 10 math - real numbers 236 , Question 10

#### Question 11:

On a square handkerchief, nine circular designs each of radius 7 cm are made (see the given figure). Find the area of the remaining portion of the handkerchief.

From the figure, it can be observed that the side of the square is 42 cm.

Area of square = (Side)2 = (42)2 = 1764 cm2

Area of each circle = πr2

Area of 9 circles = 9 × 154 = 1386 cm2

Area of the remaining portion of the handkerchief = 1764 − 1386 = 378 cm2

##### Video Solution for real numbers (Page: 237 , Q.No.: 11)

NCERT Solution for Class 10 math - real numbers 237 , Question 11

#### Question 12:

In the given figure, OACB is a quadrant of circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the

(i) Since OACB is a quadrant, it will subtend 90° angle at O.

(ii) Area of ΔOBD

Area of the shaded region = Area of quadrant OACB − Area of ΔOBD

##### Video Solution for real numbers (Page: 237 , Q.No.: 12)

NCERT Solution for Class 10 math - real numbers 237 , Question 12

#### Question 13:

In the given figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. [Use π = 3.14]

In ΔOAB,

OB2 = OA2 + AB2

= (20)2 + (20)2

Area of OABC = (Side)2 = (20)2 = 400 cm2

Area of shaded region = Area of quadrant OPBQ − Area of OABC

= (628 − 400) cm2

= 228 cm2

##### Video Solution for real numbers (Page: 237 , Q.No.: 13)

NCERT Solution for Class 10 math - real numbers 237 , Question 13

#### Question 14:

AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O (see the given figure). If ∠AOB = 30°, find the area of the shaded region.

Area of the shaded region = Area of sector OAEB − Area of sector OCFD

##### Video Solution for real numbers (Page: 237 , Q.No.: 14)

NCERT Solution for Class 10 math - real numbers 237 , Question 14

#### Question 15:

In the given figure, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

As ABC is a quadrant of the circle, ∠BAC will be of measure 90º.

In ΔABC,

BC2 = AC2 + AB2

= (14)2 + (14)2

Radius (r1) of semi-circle drawn on

Area of

Area of sector

= 154 − (154 − 98)

= 98 cm2

##### Video Solution for real numbers (Page: 237 , Q.No.: 15)

NCERT Solution for Class 10 math - real numbers 237 , Question 15

#### Question 16:

Calculate the area of the designed region in the given figure common between the two quadrants of circles of radius 8 cm each.

The designed area is the common region between two sectors BAEC and DAFC.

Area of sector

Area of ΔBAC

Area of the designed portion = 2 × (Area of segment AEC)

= 2 × (Area of sector BAEC − Area of ΔBAC)