Areas Related to Circles

Mathematics Class Tenth

10th-Math-home


NCERT Solution of Exercise 12.2

Unless stated otherwise, use 10 math area related to circle ex12.2_1

Question (1) Find the area of a sector of a circle with radius 6 cm if angle of the sector is 600.

Solution

10 math area related to circle ex12.2_1

Given, Radius of the circle = 6 cm

Angle of sector, θ = 600

Thus, Area of sector of circle = ?

We know that, area of sector of angle, θ

10 math area related to circle ex12.2_2q

Thus, area of given, sector of angle, 600

10 math area related to circle ex12.2_3q

=18.857 cm2

Thus, area of the given sector of circle = 18.857 cm2 Answer

Question (2) Find the area of a quadrant of a circle whose circumference is 22 cm.

Solution

Given, circumference = 22 cm

Area of quadrant of circle =?

We know that, circumference of a circle = 2 π r

Therefore, circumference of given circle = 2 π r

⇒ 22cm = 2 π r;

10 math area related to circle ex12.2_4q

Calculation of area of quadrant of circle

Quadrant means 1/4 part of a circle

This means angle of sector of quadrant

10 math area related to circle ex12.2_5q

Thus, angle of quadrant sector, θ = 900

Now, we know that, area of sector of angle, θ 10 math area related to circle ex12.2_2q1

Here, r = 3.5 cm and angle, θ 900

Thus, area of given quadrant of circle

10 math area related to circle ex12.2_7q

Thus, area of given quadrant = 9.625 cm2 Answer

Alternate method

Quadrant of a circle means, 1/4 part of a circle

This, means 1/4 part of area of circle will be equal to the area of quadrant of a circle.

We know that, area of a circle = π r2

Therefore, area of given circle = π (3.5 cm)2

10 math area related to circle ex12.2_8q

Now, area of Quadrant = 1/4 of Area of circle

10 math area related to circle ex12.2_9q

= 9.625 cm2 Answer

Question (3) The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minute.

solution

[strategy to solve the question (a) Find the angle made by minute hand in given time of 5 minute. (b) Calculate the area of sector of angle made by minute hand in given time.]

10 math area related to circle ex12.2_2

Given, length of the minute hand = 14 cm

This means, radius of circle = 14 cm

Area swept by given minute hand in 5 minute = ?

We know that in 60 minute hand of a clock revolve one round of the clock.

This means total angle made by minute hand = 3600

∵ In 60 minute angle formed by a minute hand = 3600

∴ In 1 minute angle formed by a minute hand

10 math area related to circle ex12.2_q10q

∴ In 5 minute angle formed by a minute hand = 60 × 5 = 300

Now, here θ = 300

And, radius, r = 14 cm

We know that, Area of sector of angle θ

10 math area related to circle ex12.2_2q12

Therefore, Area of sector of angle, 300

10 math area related to circle ex12.2_12q

= 51.33 cm 2

Thus, Area swept by given minute hand in 5 minute = 51.33 cm 2 Answer

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)

Solution

[Strategy to solve the question (a) Find the area of minor sector using radius given. (b) Find the area of right angled triangle formed by chord in the centre. (c) Subtract the area of triangle from area of minor sector, this will give the area of minor segment. (d) Find the area of circle. (e) Subtract area of minor sector from area of circle, this will give the area of major sector ]

10 math area related to circle ex12.2_3

Given, Radius of circle, r = 10 cm

Angle subtended by radius at the centre = 900

Area of minor segment =?

Area of major sector =?

We know that, area of sector of angle, θ

10 math area related to circle ex12.2_2q11

Therefore, area of sector of angle, 900

10 math area related to circle ex12.2_2q12

= 78.5 cm 2

Calculation of area of triangle

Now, area of triangle formed at the centre

Angle of triangle = 900

Height of the triangle = 10 cm (radius of circle is equal to height of the triangle)

Base of the triangle = 10 cm (radius of circle is equal to base of the triangle)

We know that Area of triangle = 1/2 × height × base

10 math area related to circle ex12.2_2q13

Thus, area of triangle formed by chord at centre = 50 cm2

Calculation of area of Minor Segment

Now, Area of minor segment, APB

= Area of sector – Area of triangle OAB

= 78.5 cm 2 – 50 cm2

= 28.5 cm2

Calculation of area of major sector

We know that, area of circle = π r2

Thus, area of given circle = π (10 cm)2

= 3.14 × 100 cm2

= 314 cm2

Now, Area of major sector = Area of circle – Area of sector

= 314 cm – 78.5 cm 2

= 235.5 cm2

Thus, Area of minor segment = 28.5 cm2 Answer

And, Area of major sector = 235.5 cm2 Answer

Question (5) In a circle of radius 21 cm, an arc subtends angle of 600 at the centre. Find

(i) The length of the arc

(ii) Area of the sector formed by the arc

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

Solution

10 math area related to circle ex12.2_4

Given, radius of chord of circle = 21cm

Angle subtended b chord = 600

(i) Calculation of length of the arc

We know that length of an arc of sector of an angle θ

10 math area related to circle ex12.2_2q14

Therefore, length of the arc of sector of given angle, 600

10 math area related to circle ex12.2_2q15

Thus, length of the arc = 22 cm

(ii) Calculation of area of sector formed by arc

We know that area of sector of angle, θ

10 math area related to circle ex12.2_2q161

Therefore, Area of sector formed by arc of angle, 600

10 math area related to circle ex12.2_2q16

= 231 cm2

Thus, Area of sector formed by arc = 231 cm2

(iii) Calculation of area of segment formed by corresponding chord

Here, triangle OAB is formed by the corresponding chord

Here, angle of triangle formed by corresponding chord = 600

And sides of triangle, which are

OA = OB = 21 cm

[Since, OA and OB are radii of the given circle]

Here, since angle of triangle is 600 and two sides are equal, thus, triangle is an equilateral triangle.

We know that, area of an equilateral triangle 10 math area related to circle ex12.2_2q17

10 math area related to circle ex12.2_2q18

= 190.953 cm2

Now, area of minor segment APB = Area of sector – Area of triangle

= 231 cm2 – 190.953 cm2

= 40.047 cm2

Thus, Answer =

Length of the arc = 22 cm

Area of sector formed by arc = 231 cm2

Area of segment formed by the corresponding chord = 40.047 cm2

Question (6) A chord of a circle of radius 15 cm subtends an angle of 600 at the centre. Find the area of the corresponding minor and major segments of the circle (Use π 3.14 and √ 3 = 1.73)

Solution

[Strategy to solve the question (a) Find the area of minor sector OAOB (b) Find area of triangle OAB formed by the given chord (c) Subtract are of triangle OAB from the area of minor sector OAPB to find the area of minor segment (d) Subtract the area of minor sector AOB from area of circle to find the area of major segment AQB]

10 math area related to circle ex12.2_5

Given, Radius of circle = 15 cm

Angle subtended by chord at centre = 600

Area of minor and major segment = ?

We know that area of sector of angle, θ 10 math area related to circle ex12.2_2q19

Therefore, Area of sector of angle, 600

10 math area related to circle ex12.2_2q19

= 117.75 cm2

Now, since angle subtended by chord at centre = 600

Thus, triangle OAB which is formed by chord is an equilateral triangle

Now, in triangle OAB

OA = OB = 15 cm, and ∠ O = 600

We know that area of an equilateral triangle 10 math area related to circle ex12.2_2q20

Thus, area of triangle OAB

10 math area related to circle ex12.2_2q20

= 97.3125 cm2

Now, area of minor segment APB

= Area of minor sector OAPB – Area of triangle OAB

= 117.75 cm2 – 97.3125 cm2

= 20.4375 cm2

We know that, area of circle = π r2

Therefore, Area of given circle = 3.14 × (15 cm)2

= 3.14 × 225 cm2

= 706.50 cm2

Now, area of major segment AQB

= Area of circle – Area of minor segment APB

= 706.50 cm2 – 20.4375 cm2

= 686.0625 cm2

Thus, Area of minor segment = 20.4375 cm2 Answer

And, Area of major segment = 686.0625 cm2 Answer

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