Areas Related to Circles

NCERT Solution of Exercise 12.2

Unless stated otherwise, use

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

Solution

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

Thus, area of given, sector of angle, 60⁰

=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;

Calculation of area of quadrant of circle

Quadrant means 1/4 part of a circle

This means angle of sector of quadrant

Thus, angle of quadrant sector, θ = 90⁰

Now, we know that, area of sector of angle, θ

Here, r = 3.5 cm and angle, θ 90⁰

Thus, area of given quadrant of circle

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

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

= 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.]

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

∴ 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

Therefore, Area of sector of angle, 30⁰

= 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 ]

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

Therefore, area of sector of angle, 90⁰

= 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

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

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

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

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

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

= 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

= 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]

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

Therefore, Area of sector of angle, 60⁰

= 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

Thus, area of triangle OAB

= 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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