8th Maths : Chapter 2 : Measurements : Parts of a Circle: Exercise 2.1 : Text Book Back Exercises Questions with Answers, Solution

**Exercise 2.1**

** **

**1. Fill in the blanks:**

(i) The ratio
between the circumference and diameter of any circle is _______. **[Answer: π ]**

(ii) A line
segment which joins any two points on a circle is a ___________. **[Answer: chord]**

(iii) The
longest chord of a circle is __________. **[Answer: diameter]**

(iv) The
radius of a circle of diameter 24 *cm* is
_______.**[Answer: 12 cm]**

(v) part
of circumference of a circle is called as _______.**[Answer: an arc]**

** **

**2. Match the following:**

(i) Area
of a circle - (a) 1/4 πr^{2}

(ii)
Circumference of a circle - (b) (π + 2)r

(iii) Area
of the sector of a circle - (c) π*r*^{2}

(iv) Circumference
of a semicircle - (d) 2 π r

(v) Area
of a quadrant of a circle - (e) θº/360° × πr^{2}

**[Answer: (i) − c, (ii) −d (iii) − e, (iv) − b, (v) – a]**

(i) Area of a circle **c. π
r^{2}**

(ii) Circumference of a circle **d. 2π r**

(iii) Area of the sector of a circle **e. [θ° / 360°] × π r^{2}**

(iv) Circumference of a semicircle **b. (π + 2) r**

(v) Area of a quadrant of a circle **a. 1/4 π r^{2}**

**3. Find the central angle of the shaded
sectors (each circle is divided into equal sectors).**

Solution:

** **

**4. For the sectors with given measures,
find the length of the arc, area and perimeter. (π=3.14)**

**(i) central angle 45º, r = 16 cm (ii) central angle 120º, d
=12.6 cm **

Solution:

**(i) Central angle
45°, r = 16 cm **

Length of the arc *l* = [ θ° / 360° ] × 2π*r* units

*l* = [ 45° / 360° ] × 2 × 3.14 × 16 cm

*l* = 1/8 × 2 × 3.14 × 16 cm

*l* = 12.56 cm

Area of the sector = [ θ° / 360° ] × π*r*^{2} sq.units

A = [ 45° / 360° ] × 3.14
× 16 × 16

A = 100.48 cm^{2}

Perimeter of the sector P = *l* + 2*r* units

P = 12.56 + 2(16) cm

P = 44.56 cm

**(ii) central angle
120°, d =12.6 cm**

∴ *r* = 12.6 / 2 cm

*r* = 6.3 cm

Length of the arc *l* = [ θ° / 360° ] × 2 π*r* units

*l* = [ 120° / 360° ] × 2 × 3.14 × 6.3
cm

*l* = 13.188 cm

*l* = 13.19 cm

Area of the sector A = [ θ°
/ 360° ] × π*r*^{2} sq.units

A = [ 120° / 360° ] × 3.14
× 6.3 × 6.3 cm^{2}

A = 3.14 × 6.3 × 2.1 cm^{2}

A = 41.54 cm^{2}

Perimeter of the sector P = *l* + 2*r* cm

P = 13.19 + 2(6.3) cm

P = 13.19 + 12.6 cm

P = 25.79 cm

** **

**5. From the measures given below, find
the area of the sectors.**

**(i) length of the arc = 48 m, r
= 10 m (ii) length of the arc = 50 cm, r
= 13.5 cm**

**Solution:**

**(i) Length of the arc = 48 m, r = 10 m**

Area of the sector A = *lr*/2 sq. units

*l* = 48 m

*r* = 10 m

= [ 48 × 10 ]/2 m^{2}

= 24 × 10 m^{2}

= 240 m^{2}

Area of the sector = 240 m^{2}

**(ii) Length of the arc = 50 cm, r = 13.5 cm**

Length of the arc *l* = 50 cm

Radius *r* = 13.5 cm

Area of the sector A = (*lr*/2) sq. units

A = [50 × 13.5] / 2

A = 25 × 13.5 cm

A = 337.5 cm

Area of the sector A = 337.5 cm^{2}

** **

**6. Find the central angle of each of
the sectors whose measures are given below. (***π*** ****=**** 22/**** 7)**

**(i) area = 462 cm^{2},
r = 21 cm (ii) length of the arc = 44
m, r = 35 m**

**Solution: **

**(i) area = 462 cm ^{2}, r = 21 cm**

Radius of the sector = 21 cm

Area of the sector = 462 cm^{2}

* lr */ 2 = 462

* *[ *l* × 21] / 2 = 462

*l* = [ 462 × 2 ] / 21

*l* = 22 × 2

Length of the arc *l* = 44 cm

[ θ° / 360° ] × 2π*r* = 44 cm

[θ° / 360°] × 2 × [22/7] × 21 = 44 cm

θ° = [ 44 × 360 × 7 ] / [ 2 × 22 × 21 ]

θ° = 120°

∴ Central angle of the sector = 120°.

**(ii) length of the arc = 44 m, r = 35 m **

Length of the arc = 44 cm

*r* = 35 cm

[θ° / 360°] × 2π*r* = 44 cm

[θ° / 360] × 2 × [22/7] × 35 = 44 cm

θ° = [ 44 × 360 × 7] / [ 2 × 22 × 35]

= 72°

Central angle = 72°

** **

**7. A circle of radius 120 m is
divided into 8 equal sectors. Find the length of the arc of each of the sectors.**

**Solution:**

Radius of the circle *r* = 120 m

Number of equal sectors = 8

∴ Central angle of each sector = 360° / *n*

θ° = 360° / 8

θ° = 45°

Length of the arc *l* = [ θ° / 360°] × 2π*r* units

= [45° / 360°] × 2π × 120 m

Length of the arc = 30 × π m

**Another method:**

*l* = [ 1/*n* ] × 2π*r* = [1/8]
× 2 × π × 120 = 30 π m

Length of the arc = 30 π m

** **

**8. A circle of radius 70 cm is
divided into 5 equal sectors. Find the area of each of the sectors.**

**Solution:**

Radius of the sector *r *= 70 cm

Number of equal sectors = 5

∴ Central angle of each sector = 360° / *n*

θ° = 360° / 5

θ° = 72°

Area of the sector = [θ° / 360°] × π*r*^{2}
sq.units

= [72° / 360°] × π × 70 × 70 cm^{2}

= 14 × 70 × π *cm*^{2}

= 980 π cm^{2}

Note : We can solve this problem using A = (1/*n*) π*r*^{2 }sq. units
also.

** **

**9. Dhamu fixes a square tile of 30 cm
on the floor. The tile has a sector design on it as shown in the figure. Find the
area of the sector. (π = 3.14) .**

**Solution:**

Side of the square = 30 cm

∴ Radius of the sector design = 30 cm

Given the design of a circular quadrant.

Area of the quadrant = (1/4) × π*r*^{2} sq.units

= (1/4) × 3.14 × 30 × 30
cm^{2}

= 3.14 × 15 × 15 cm^{2}

∴ Area of the sector design = 706.5 cm^{2} (approximately)

** **

**10. A circle is formed with 8 equal granite stones as shown in the figure
each of radius 56 cm and whose central
angle is 45º. Find the area of each of the granite stones. (π = 22/7)**

**Solution:**

Number of equal sectors ‘*n*’ = 8

Radius of the sector ‘*r*’ = 56 cm

Area of each sector = (1/*n*) π*r*^{2} sq.units

= (1/8) × (22/7) × 56 × 56 cm^{2} = 1232 cm^{2}

Area of each sector = 1232 cm^{2} (approximately)

** **

**Answer:**

**Exercise
2.1 **

**1. (i) π (ii) chord (iii)
diameter (iv) 12 cm (v) circular arc **

**2. (i) c (ii) d (iii) e
(iv) b (v) a **

**3. **

**4. **

**5. (i) 240 m^{2} (ii) 337.5 cm^{2} **

**6. (i) θ = 120º (ii) θ
= 72º **

**7. 30 π m**

**8. 980***π*** cm^{2}**

**9. 706.5 cm^{2} (approximately)**

**10. 1232 cm^{2} (approximately)**

Tags : Questions with Answers, Solution | Measurements | Chapter 2 | 8th Maths , 8th Maths : Chapter 2 : Measurements

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8th Maths : Chapter 2 : Measurements : Exercise 2.1 (Parts of a Circle) | Questions with Answers, Solution | Measurements | Chapter 2 | 8th Maths

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