##### Class 9^{th} Mathematics Term 3 Tamilnadu Board Solution

**Exercise 4.1**- radius 21 cm and central angle 60° Find the arc length, area and perimeter of…
- radius 4.9 cm and central angle 30° Find the arc length, area and perimeter of…
- radius 14 cm and sector angle 45° Find the arc length, area and perimeter of…
- radius 15 cm and sector angle 63° Find the arc length, area and perimeter of…
- radius 21 dm and sector angle 240° Find the arc length, area and perimeter of…
- Find the angle subtended by an arc 88 cm long at the centre of a circle of…
- The arc length of a sector of a circle of radius 14 cm is 22 cm. Find its…
- Find the radius of a sector of a circle having a central angle 70°and an arc…
- radius 10 cm and arc length 33 cm. Find the area and perimeter of the sector…
- radius 55 cm and arc length 80 cm. Find the area and perimeter of the sector…
- radius 12 cm and arc length 15.25 cm. Find the area and perimeter of the sector…
- radius 20 cm and arc length 25 cm. Find the area and perimeter of the sector…
- Find the arc length of the sector of radius 14 cm and area 70 cm^2…
- Find the radius of the sector of area 225 cm^2 and having an arc length of 15…
- Find the radius of the sector whose central angle is 140°and area 44 cm^2 .…
- The perimeter of a sector of a circle is 58 cm. Find the area if its diameter…
- Find the area of a sector whose radius and perimeter are 20 cm and 110 cm…
- area 352 cm^2 and radius 12 cm Find the central angle of a sector of a circle…
- area 462 cm^2 and radius 21 cm Find the central angle of a sector of a circle…
- Calculate the perimeter and area of the semicircle whose radius is 14 cm.…
- Calculate the perimeter and area of a quadrant circle of radius 7 cm.…
- Calculate the arc length of a sector whose perimeter and radius are 35 cm and 8…
- Find the radius of a sector whose perimeter and arc length are 24 cm and 7 cm…
- Time spent by a student in a day is shown in the figure. Find how much time is…
- Three coins each 2 cm in diameter are placed touching one another. Find the…
- Four horses are tethered with ropes measuring 7 m each to the four corners of a…
- Find the area of card board wasted if a sector of maximum possible size is cut…
- Find the area of the shaded portion in the adjoining figure
- Find the radius, central angle and perimeter of a sector whose length of arc…

**Exercise 4.2**- 5.6 cm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 6 dm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 2.5 m Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 24 cm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 31 cm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- If the Lateral Surface Area of a cube is 900 cm^2 , find the length of its…
- If the Total Surface Area of a cube is 1014 cm^2 , find the length of its side.…
- The volume of the cube is 125 dm^3 . Find its side.
- A container is in the shape of a cube of side 20 cm. How much sugar can it hold?…
- A cubical tank can hold 64,000 litres of water. Find the length of the side of…
- Three metallic cubes of side 3 cm, 4 cm and 5 cm respectively are melted and are…
- How many cubes of side 3 cm are required to build a cube of side 15 cm?…
- Find the area of card board required to make an open cubical box of side 40 cm.…
- What is the total cost of oil in a cubical container of side 2 m if it is…
- A container of side 3.5m is to be painted both inside and outside. Find the area…

**Exercise 4.3**- respectively as 5 cm, 2 cm , 11cm Find the L.S.A, T.S.A and volume of the…
- respectively as 15 dm, 10 dm, 8 dm Find the L.S.A, T.S.A and volume of the…
- respectively as 2 m, 3 m, 7 m Find the L.S.A, T.S.A and volume of the cuboids…
- respectively as 20 m, 12 m, 8 m Find the L.S.A, T.S.A and volume of the cuboids…
- Find the height of the cuboid whose length, breadth and volume are 35 cm, 15 cm…
- Two cubes each of volume 64 cm^3 are joined to form a cuboid. Find the L.S.A and…
- Raju planned to stitch a cover for his two speaker boxes whose length, breadth…
- Mohan wanted to paint the walls and ceiling of a hall. The dimensions of the…
- How many hollow blocks of size 30cm x 15cm x20cm are needed to construct a wall…
- Find the cost of renovating the walls and the floor of a hall that measures 10m…

**Exercise 4.4**- The length of the arc of a sector having central angle 90° and radius 7 cm isA.…
- If the radius and arc length of a sector are 17 cm and 27 cm respectively, then…
- If the angle subtended by the arc of a sector at the center is 90°, then the…
- Area of a sector having radius 12 cm and arc length 21 cm isA. 126 cm^2 B. 252…
- The area of a sector with radius 4 cm and central angle 60° isA. 2 pi /3 cm^2 B.…
- If the area and arc length of the sector of a circle are 60 cm^2 and 20 cm…
- The perimeter of a sector of a circle is 37cm. If its radius is 7cm, then its…
- A solid having six equal square faces is called aA. cube B. cuboid C. square D.…
- The quantity of space occupied by a body is itsA. area B. length C. volume D.…
- The LSA of a cube of side 1dm isA. 16 dm^2 B. 4 dm^2 C. 2 dm^2 D. 1 dm^2…

**Exercise 4.1**

- radius 21 cm and central angle 60° Find the arc length, area and perimeter of…
- radius 4.9 cm and central angle 30° Find the arc length, area and perimeter of…
- radius 14 cm and sector angle 45° Find the arc length, area and perimeter of…
- radius 15 cm and sector angle 63° Find the arc length, area and perimeter of…
- radius 21 dm and sector angle 240° Find the arc length, area and perimeter of…
- Find the angle subtended by an arc 88 cm long at the centre of a circle of…
- The arc length of a sector of a circle of radius 14 cm is 22 cm. Find its…
- Find the radius of a sector of a circle having a central angle 70°and an arc…
- radius 10 cm and arc length 33 cm. Find the area and perimeter of the sector…
- radius 55 cm and arc length 80 cm. Find the area and perimeter of the sector…
- radius 12 cm and arc length 15.25 cm. Find the area and perimeter of the sector…
- radius 20 cm and arc length 25 cm. Find the area and perimeter of the sector…
- Find the arc length of the sector of radius 14 cm and area 70 cm^2…
- Find the radius of the sector of area 225 cm^2 and having an arc length of 15…
- Find the radius of the sector whose central angle is 140°and area 44 cm^2 .…
- The perimeter of a sector of a circle is 58 cm. Find the area if its diameter…
- Find the area of a sector whose radius and perimeter are 20 cm and 110 cm…
- area 352 cm^2 and radius 12 cm Find the central angle of a sector of a circle…
- area 462 cm^2 and radius 21 cm Find the central angle of a sector of a circle…
- Calculate the perimeter and area of the semicircle whose radius is 14 cm.…
- Calculate the perimeter and area of a quadrant circle of radius 7 cm.…
- Calculate the arc length of a sector whose perimeter and radius are 35 cm and 8…
- Find the radius of a sector whose perimeter and arc length are 24 cm and 7 cm…
- Time spent by a student in a day is shown in the figure. Find how much time is…
- Three coins each 2 cm in diameter are placed touching one another. Find the…
- Four horses are tethered with ropes measuring 7 m each to the four corners of a…
- Find the area of card board wasted if a sector of maximum possible size is cut…
- Find the area of the shaded portion in the adjoining figure
- Find the radius, central angle and perimeter of a sector whose length of arc…

**Exercise 4.2**

- 5.6 cm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 6 dm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 2.5 m Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 24 cm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- 31 cm Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume…
- If the Lateral Surface Area of a cube is 900 cm^2 , find the length of its…
- If the Total Surface Area of a cube is 1014 cm^2 , find the length of its side.…
- The volume of the cube is 125 dm^3 . Find its side.
- A container is in the shape of a cube of side 20 cm. How much sugar can it hold?…
- A cubical tank can hold 64,000 litres of water. Find the length of the side of…
- Three metallic cubes of side 3 cm, 4 cm and 5 cm respectively are melted and are…
- How many cubes of side 3 cm are required to build a cube of side 15 cm?…
- Find the area of card board required to make an open cubical box of side 40 cm.…
- What is the total cost of oil in a cubical container of side 2 m if it is…
- A container of side 3.5m is to be painted both inside and outside. Find the area…

**Exercise 4.3**

- respectively as 5 cm, 2 cm , 11cm Find the L.S.A, T.S.A and volume of the…
- respectively as 15 dm, 10 dm, 8 dm Find the L.S.A, T.S.A and volume of the…
- respectively as 2 m, 3 m, 7 m Find the L.S.A, T.S.A and volume of the cuboids…
- respectively as 20 m, 12 m, 8 m Find the L.S.A, T.S.A and volume of the cuboids…
- Find the height of the cuboid whose length, breadth and volume are 35 cm, 15 cm…
- Two cubes each of volume 64 cm^3 are joined to form a cuboid. Find the L.S.A and…
- Raju planned to stitch a cover for his two speaker boxes whose length, breadth…
- Mohan wanted to paint the walls and ceiling of a hall. The dimensions of the…
- How many hollow blocks of size 30cm x 15cm x20cm are needed to construct a wall…
- Find the cost of renovating the walls and the floor of a hall that measures 10m…

**Exercise 4.4**

- The length of the arc of a sector having central angle 90° and radius 7 cm isA.…
- If the radius and arc length of a sector are 17 cm and 27 cm respectively, then…
- If the angle subtended by the arc of a sector at the center is 90°, then the…
- Area of a sector having radius 12 cm and arc length 21 cm isA. 126 cm^2 B. 252…
- The area of a sector with radius 4 cm and central angle 60° isA. 2 pi /3 cm^2 B.…
- If the area and arc length of the sector of a circle are 60 cm^2 and 20 cm…
- The perimeter of a sector of a circle is 37cm. If its radius is 7cm, then its…
- A solid having six equal square faces is called aA. cube B. cuboid C. square D.…
- The quantity of space occupied by a body is itsA. area B. length C. volume D.…
- The LSA of a cube of side 1dm isA. 16 dm^2 B. 4 dm^2 C. 2 dm^2 D. 1 dm^2…

###### Exercise 4.1

**Question 1.**Find the arc length, area and perimeter of the sector with

radius 21 cm and central angle 60°

**Answer:**Given:

radius (r) = 21 cm

central angle 60°

Arc length = ?, area of sector = ? and perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 22cm

(b) Area of sector =

⇒ area of sector = 231cm^{2}

(c) perimeter of sector = r + r + length of arc

= 2(r) + length of arc

= 2(21) + 22

= 42 + 22 = 64cm

**Question 2.**Find the arc length, area and perimeter of the sector with

radius 4.9 cm and central angle 30°

**Answer:**Given:

radius (r) = 4.9 cm

central angle 30°

Arc length = ?, area of sector = ? and perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 2.57cm

(b) Area of sector =

⇒ area of sector = 6.3cm^{2}

( c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(4.9) + 2.57

= 9.8 + 2.57 = 12.37cm

**Question 3.**Find the arc length, area and perimeter of the sector with

radius 14 cm and sector angle 45°

**Answer:**Given:

radius (r) = 14 cm

central angle 45°

Arc length = ?, area of sector = ? and perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 11cm

(b) Area of sector =

⇒ area of sector = 77cm^{2}

(c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(14) + 11

= 28 + 11 = 39cm

**Question 4.**Find the arc length, area and perimeter of the sector with

radius 15 cm and sector angle 63°

**Answer:**Given:

radius (r) = 15 cm

central angle 63°

Arc length = ?

area of sector = ?

perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 16.5cm

(b) Area of sector =

⇒ area of sector = 123.75cm^{2}

(c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(15) + 16.5

= 30 + 16.5 = 46.5cm

**Question 5.**Find the arc length, area and perimeter of the sector with

radius 21 dm and sector angle 240°

**Answer:**Given:

radius (r) = 21 dm

central angle 240°

Arc length = ?

area of sector = ?

perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 88dm

(b) Area of sector =

⇒ area of sector = 924 dm^{2}

(c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(21) + 88

= 42 + 88 = 130dm

**Question 6.**Find the angle subtended by an arc 88 cm long at the centre of a circle of radius 42 cm.

**Answer:**Given:

radius (r) = 42 cm

Arc length = 88 cm

central angle = ?

As we know,

Length of arc =

⇒ central angle = 12°

**Question 7.**The arc length of a sector of a circle of radius 14 cm is 22 cm. Find its central angle.

**Answer:**Given:

radius (r) = 14 cm

Arc length = 22 cm

central angle = θ

As we know,

Length of arc =

⇒ central angle = 90°

**Question 8.**Find the radius of a sector of a circle having a central angle 70°and an arc length of 44 cm.

**Answer:**Given:

radius (r) = ? cm

Arc length = 44 cm

central angle = 70°

As we know,

Length of arc =

⇒ radius = 36cm

**Question 9.**Find the area and perimeter of the sector with

radius 10 cm and arc length 33 cm.

**Answer:**Given:

radius (r) = 10 cm

Arc length (a) = 33cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 165 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(10) + 33

= 20 + 33 = 53cm

**Question 10.**Find the area and perimeter of the sector with

radius 55 cm and arc length 80 cm.

**Answer:**Given:

radius (r) = 55 cm

Arc length (a) = 80cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 2200 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(55) + 80

= 110 + 80 = 190cm

**Question 11.**Find the area and perimeter of the sector with

radius 12 cm and arc length 15.25 cm.

**Answer:**Given:

radius (r) = 12 cm

Arc length (a) = 15.25cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 91.5 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(12) + 15.25

= 24 + 15.25 = 39.25cm

**Question 12.**Find the area and perimeter of the sector with

radius 20 cm and arc length 25 cm.

**Answer:**Given: radius (r) = 20 cm

Arc length (a) = 25cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 250 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(20) + 25

= 40 + 25 = 65cm

**Question 13.**Find the arc length of the sector of radius 14 cm and area 70 cm^{2}

**Answer:**Given:

radius (r) = 14 cm

Arc length (a) = ?

area of sector = 70cm^{2}

As we know,

Area of sector =

⇒ arc length = 10 cm

**Question 14.**Find the radius of the sector of area 225 cm^{2} and having an arc length of 15 cm

**Answer:**Given:

radius (r) = ?

Arc length (a) = 15cm

area of sector = 225cm^{2}

As we know,

Area of sector =

⇒ radius = 30 cm

**Question 15.**Find the radius of the sector whose central angle is 140°and area 44 cm^{2}.

**Answer:**Given:

radius (r) = ?

central angle (a) = 140°

area of sector = 44cm^{2}

Area of sector =

⇒ r^{2} = 36

⇒ r = 6 cm

**Question 16.**The perimeter of a sector of a circle is 58 cm. Find the area if its diameter is 9 cm.

**Answer:**perimeter of a sector = 58 cm

diameter = 9 cm

Radius (r) = d/2 = 9/2 = 4.5 cm

Perimeter = r + r + length of arc

= 2 ( r) + length of arc

58 =arc length + 9

Arc length (a)= 58 – 9 = 49cm

Area of sector =

⇒ area of sector = 110.25 cm^{2}

**Question 17.**Find the area of a sector whose radius and perimeter are 20 cm and 110 cm respectively.

**Answer:**perimeter of a sector = 110 cm

radius = 20 cm

Perimeter = r + r + length of arc

= 2 ( r) + length of arc

110 =arc length + 2(20)

Arc length (a)= 110 – 40 = 70cm

Area of sector =

⇒ area of sector = 700 cm^{2}

**Question 18.**Find the central angle of a sector of a circle having

area 352 cm^{2} and radius 12 cm

**Answer:**Given:

radius (r) = 12 cm

central angle = ?

area of sector = 352 cm^{2}

As we know,

Area of sector =

⇒ central angle = 280°

**Question 19.**Find the central angle of a sector of a circle having

area 462 cm^{2} and radius 21 cm

**Answer:**Given:

radius (r) = 21 cm

central angle = ?

area of sector = 462 cm^{2}

As we know,

Area of sector =

⇒ central angle = 120°

**Question 20.**Calculate the perimeter and area of the semicircle whose radius is 14 cm.

**Answer:**Given:

radius (r)= 14cm

(a) perimeter of semi-circle = 1/2 (2πr) + r + r

= π r + 2r

perimeter of semi-circle = 44 + 28 = 72cm

(b) area of semi-circle =

area of semi-circle = 308 cm^{2}

**Question 21.**Calculate the perimeter and area of a quadrant circle of radius 7 cm.

**Answer:**Given:

radius (r)= 7cm

(a) perimeter of quadrant-circle = 1/4 (2 π r) + r + r

= 1/2 π r + 2r

perimeter of quadrant-circle = 11 + 14 = 25cm

(b) area of quadrant-circle =

area of quadrant-circle = 38.5 cm^{2}

**Question 22.**Calculate the arc length of a sector whose perimeter and radius are 35 cm and 8 cm respectively.

**Answer:**perimeter of a sector = 35 cm

radius = 8 cm

Perimeter = arc length + r + r

= arc length + 2r

35 =arc length + 2(8)

Arc length (a)= 35 – 16 = 19cm

**Question 23.**Find the radius of a sector whose perimeter and arc length are 24 cm and 7 cm respectively.

**Answer:**perimeter of a sector = 24 cm

arc length = 7 cm

Perimeter = arc length + r + r

= arc length + 2r

24 = 7 + 2(r)

2r = 24 – 7

r = 17/2

r = 8.5 cm

**Question 24.**Time spent by a student in a day is shown in the figure. Find how much time is spent in

(i) school (ii) play ground (iii) other activities

**Answer:**Given:

Total time available is 24hr

(i) time spent in school =

time spent in school = 7hr

(ii) time spent in playground =

time spent in playground = 2hr

(iii) time spent in other activities =

time spent in other activities = 15hr

**Question 25.**Three coins each 2 cm in diameter are placed touching one another. Find the area enclosed by them.

**Answer:**Given:

diameter of each coin = 2cm

⇒ Radius(r) = 1cm

Area enclosed by one coin = π r^{2}

=

Area enclosed by all the 3 coin = 3× 22/7

= 9.43cm^{2}

**Question 26.**Four horses are tethered with ropes measuring 7 m each to the four corners of a rectangular grass land 21 m x 24 m in dimension. Find

(i) the maximum area that can be grazed by the horses

(ii) the area that remains ungrazed.

**Answer:**Given:

(i) length of rope = radius of circle.

At one corner area grazed =

At one corner area grazed = 38.5m

the maximum area that can be grazed by the horses = 4 × area at one corner

= 4 × 38.5

= 154m^{2}

(ii) Area that remain ungrazed = total area of rectangle – area grazed by horses

Area that remain ungrazed = (21 m x 24 m) – 154

= 504 – 154

= 350 m^{2}

**Question 27.**Find the area of card board wasted if a sector of maximum possible size is cut out from a square card board of size 24 cm. [π = 3.14].

**Answer:**Given:

The area of square = (side)^{2} = 24× 24 cm^{2} = 576 cm^{2}

Now, the area of area has to be maximum. It means that the side of the square will be radius of the sector.

Then, area of the sector =

=

= 452.57 cm^{2}

Then, the wasted area will be given as the remaining area

= Area of square – Area of the sector

= 576 cm^{2} – 452.57 cm^{2}

123.84 cm^{2}

**Question 28.**Find the area of the shaded portion in the adjoining figure

**Answer:**Given:

θ = 30°

radius of small part is 21cm

radius of bigger part is 42cm

Area of small sector =

⇒ area of small sector = 115.5 cm^{2}

Area of large sector =

⇒ area of large sector = 462 cm^{2}

Area of shaded part = 462 – 115.5

= 346.5cm^{2}

**Question 29.**Find the radius, central angle and perimeter of a sector whose length of arc and area are 4.4 m and 9.24 m^{2} respectively.

**Answer:**Given:

length of arc (a) = 4.4m

area = 9.24m^{2}

(a) Area of sector =

⇒ radius of sector = 4.2m

(b) Area of sector =

⇒ central angle = 60°

(c) perimeter of sector = 2 ( r) + length of arc

= 2(4.2) + 4.4

= 8.4 + 4.4

= 12.8cm

**Question 1.**

Find the arc length, area and perimeter of the sector with

radius 21 cm and central angle 60°

**Answer:**

Given:

radius (r) = 21 cm

central angle 60°

Arc length = ?, area of sector = ? and perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 22cm

(b) Area of sector =

⇒ area of sector = 231cm^{2}

(c) perimeter of sector = r + r + length of arc

= 2(r) + length of arc

= 2(21) + 22

= 42 + 22 = 64cm

**Question 2.**

Find the arc length, area and perimeter of the sector with

radius 4.9 cm and central angle 30°

**Answer:**

Given:

radius (r) = 4.9 cm

central angle 30°

Arc length = ?, area of sector = ? and perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 2.57cm

(b) Area of sector =

⇒ area of sector = 6.3cm^{2}

( c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(4.9) + 2.57

= 9.8 + 2.57 = 12.37cm

**Question 3.**

Find the arc length, area and perimeter of the sector with

radius 14 cm and sector angle 45°

**Answer:**

Given:

radius (r) = 14 cm

central angle 45°

Arc length = ?, area of sector = ? and perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 11cm

(b) Area of sector =

⇒ area of sector = 77cm^{2}

(c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(14) + 11

= 28 + 11 = 39cm

**Question 4.**

Find the arc length, area and perimeter of the sector with

radius 15 cm and sector angle 63°

**Answer:**

Given:

radius (r) = 15 cm

central angle 63°

Arc length = ?

area of sector = ?

perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 16.5cm

(b) Area of sector =

⇒ area of sector = 123.75cm^{2}

(c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(15) + 16.5

= 30 + 16.5 = 46.5cm

**Question 5.**

Find the arc length, area and perimeter of the sector with

radius 21 dm and sector angle 240°

**Answer:**

Given:

radius (r) = 21 dm

central angle 240°

Arc length = ?

area of sector = ?

perimeter of sector = ?

As we know,

(a) Length of arc =

⇒ Length of arc = 88dm

(b) Area of sector =

⇒ area of sector = 924 dm^{2}

(c) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(21) + 88

= 42 + 88 = 130dm

**Question 6.**

Find the angle subtended by an arc 88 cm long at the centre of a circle of radius 42 cm.

**Answer:**

Given:

radius (r) = 42 cm

Arc length = 88 cm

central angle = ?

As we know,

Length of arc =

⇒ central angle = 12°

**Question 7.**

The arc length of a sector of a circle of radius 14 cm is 22 cm. Find its central angle.

**Answer:**

Given:

radius (r) = 14 cm

Arc length = 22 cm

central angle = θ

As we know,

Length of arc =

⇒ central angle = 90°

**Question 8.**

Find the radius of a sector of a circle having a central angle 70°and an arc length of 44 cm.

**Answer:**

Given:

radius (r) = ? cm

Arc length = 44 cm

central angle = 70°

As we know,

Length of arc =

⇒ radius = 36cm

**Question 9.**

Find the area and perimeter of the sector with

radius 10 cm and arc length 33 cm.

**Answer:**

Given:

radius (r) = 10 cm

Arc length (a) = 33cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 165 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(10) + 33

= 20 + 33 = 53cm

**Question 10.**

Find the area and perimeter of the sector with

radius 55 cm and arc length 80 cm.

**Answer:**

Given:

radius (r) = 55 cm

Arc length (a) = 80cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 2200 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(55) + 80

= 110 + 80 = 190cm

**Question 11.**

Find the area and perimeter of the sector with

radius 12 cm and arc length 15.25 cm.

**Answer:**

Given:

radius (r) = 12 cm

Arc length (a) = 15.25cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 91.5 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(12) + 15.25

= 24 + 15.25 = 39.25cm

**Question 12.**

Find the area and perimeter of the sector with

radius 20 cm and arc length 25 cm.

**Answer:**

Given: radius (r) = 20 cm

Arc length (a) = 25cm

area of sector = ?

perimeter of sector = ?

As we know,

(a) Area of sector =

⇒ area of sector = 250 cm^{2}

(b) perimeter of sector = r + r + length of arc

= 2 ( r) + length of arc

= 2(20) + 25

= 40 + 25 = 65cm

**Question 13.**

Find the arc length of the sector of radius 14 cm and area 70 cm^{2}

**Answer:**

Given:

radius (r) = 14 cm

Arc length (a) = ?

area of sector = 70cm^{2}

As we know,

Area of sector =

⇒ arc length = 10 cm

**Question 14.**

Find the radius of the sector of area 225 cm^{2} and having an arc length of 15 cm

**Answer:**

Given:

radius (r) = ?

Arc length (a) = 15cm

area of sector = 225cm^{2}

As we know,

Area of sector =

⇒ radius = 30 cm

**Question 15.**

Find the radius of the sector whose central angle is 140°and area 44 cm^{2}.

**Answer:**

Given:

radius (r) = ?

central angle (a) = 140°

area of sector = 44cm^{2}

Area of sector =

⇒ r^{2} = 36

⇒ r = 6 cm

**Question 16.**

The perimeter of a sector of a circle is 58 cm. Find the area if its diameter is 9 cm.

**Answer:**

perimeter of a sector = 58 cm

diameter = 9 cm

Radius (r) = d/2 = 9/2 = 4.5 cm

Perimeter = r + r + length of arc

= 2 ( r) + length of arc

58 =arc length + 9

Arc length (a)= 58 – 9 = 49cm

Area of sector =

⇒ area of sector = 110.25 cm^{2}

**Question 17.**

Find the area of a sector whose radius and perimeter are 20 cm and 110 cm respectively.

**Answer:**

perimeter of a sector = 110 cm

radius = 20 cm

Perimeter = r + r + length of arc

= 2 ( r) + length of arc

110 =arc length + 2(20)

Arc length (a)= 110 – 40 = 70cm

Area of sector =

⇒ area of sector = 700 cm^{2}

**Question 18.**

Find the central angle of a sector of a circle having

area 352 cm^{2} and radius 12 cm

**Answer:**

Given:

radius (r) = 12 cm

central angle = ?

area of sector = 352 cm^{2}

As we know,

Area of sector =

⇒ central angle = 280°

**Question 19.**

Find the central angle of a sector of a circle having

area 462 cm^{2} and radius 21 cm

**Answer:**

Given:

radius (r) = 21 cm

central angle = ?

area of sector = 462 cm^{2}

As we know,

Area of sector =

⇒ central angle = 120°

**Question 20.**

Calculate the perimeter and area of the semicircle whose radius is 14 cm.

**Answer:**

Given:

radius (r)= 14cm

(a) perimeter of semi-circle = 1/2 (2πr) + r + r

= π r + 2r

perimeter of semi-circle = 44 + 28 = 72cm

(b) area of semi-circle =

area of semi-circle = 308 cm^{2}

**Question 21.**

Calculate the perimeter and area of a quadrant circle of radius 7 cm.

**Answer:**

Given:

radius (r)= 7cm

(a) perimeter of quadrant-circle = 1/4 (2 π r) + r + r

= 1/2 π r + 2r

perimeter of quadrant-circle = 11 + 14 = 25cm

(b) area of quadrant-circle =

area of quadrant-circle = 38.5 cm^{2}

**Question 22.**

Calculate the arc length of a sector whose perimeter and radius are 35 cm and 8 cm respectively.

**Answer:**

perimeter of a sector = 35 cm

radius = 8 cm

Perimeter = arc length + r + r

= arc length + 2r

35 =arc length + 2(8)

Arc length (a)= 35 – 16 = 19cm

**Question 23.**

Find the radius of a sector whose perimeter and arc length are 24 cm and 7 cm respectively.

**Answer:**

perimeter of a sector = 24 cm

arc length = 7 cm

Perimeter = arc length + r + r

= arc length + 2r

24 = 7 + 2(r)

2r = 24 – 7

r = 17/2

r = 8.5 cm

**Question 24.**

Time spent by a student in a day is shown in the figure. Find how much time is spent in

(i) school (ii) play ground (iii) other activities

**Answer:**

Given:

Total time available is 24hr

(i) time spent in school =

time spent in school = 7hr

(ii) time spent in playground =

time spent in playground = 2hr

(iii) time spent in other activities =

time spent in other activities = 15hr

**Question 25.**

Three coins each 2 cm in diameter are placed touching one another. Find the area enclosed by them.

**Answer:**

Given:

diameter of each coin = 2cm

⇒ Radius(r) = 1cm

Area enclosed by one coin = π r^{2}

=

Area enclosed by all the 3 coin = 3× 22/7

= 9.43cm^{2}

**Question 26.**

Four horses are tethered with ropes measuring 7 m each to the four corners of a rectangular grass land 21 m x 24 m in dimension. Find

(i) the maximum area that can be grazed by the horses

(ii) the area that remains ungrazed.

**Answer:**

Given:

(i) length of rope = radius of circle.

At one corner area grazed =

At one corner area grazed = 38.5m

the maximum area that can be grazed by the horses = 4 × area at one corner

= 4 × 38.5

= 154m^{2}

(ii) Area that remain ungrazed = total area of rectangle – area grazed by horses

Area that remain ungrazed = (21 m x 24 m) – 154

= 504 – 154

= 350 m^{2}

**Question 27.**

Find the area of card board wasted if a sector of maximum possible size is cut out from a square card board of size 24 cm. [π = 3.14].

**Answer:**

Given:

The area of square = (side)^{2} = 24× 24 cm^{2} = 576 cm^{2}

Now, the area of area has to be maximum. It means that the side of the square will be radius of the sector.

Then, area of the sector =

=

= 452.57 cm^{2}

Then, the wasted area will be given as the remaining area

= Area of square – Area of the sector

= 576 cm^{2} – 452.57 cm^{2}

123.84 cm^{2}

**Question 28.**

Find the area of the shaded portion in the adjoining figure

**Answer:**

Given:

θ = 30°

radius of small part is 21cm

radius of bigger part is 42cm

Area of small sector =

⇒ area of small sector = 115.5 cm^{2}

Area of large sector =

⇒ area of large sector = 462 cm^{2}

Area of shaded part = 462 – 115.5

= 346.5cm^{2}

**Question 29.**

Find the radius, central angle and perimeter of a sector whose length of arc and area are 4.4 m and 9.24 m^{2} respectively.

**Answer:**

Given:

length of arc (a) = 4.4m

area = 9.24m^{2}

(a) Area of sector =

⇒ radius of sector = 4.2m

(b) Area of sector =

⇒ central angle = 60°

(c) perimeter of sector = 2 ( r) + length of arc

= 2(4.2) + 4.4

= 8.4 + 4.4

= 12.8cm

###### Exercise 4.2

**Question 1.**Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

5.6 cm

**Answer:**Given:

side of cube (a) = 5.6cm

(a) LSA of cube = 4a^{2}

= 4(5.6)^{2}

⇒ LSA of cube = 125.44cm^{2}

(b) TSA of cube = 6a^{2}

= 6(5.6)^{2}

⇒ TSA of cube = 188.16cm^{2}

(c) Volume of cube = a^{3}

= (5.6)^{3}

⇒ Volume of cube = 175.62cm^{3}

**Question 2.**Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

6 dm

**Answer:**Given:

side of cube (a) = 6dm

(a) LSA of cube = 4a^{2}

= 4(6)^{2}

⇒ LSA of cube = 144dm^{2}

(b) TSA of cube = 6a^{2}

= 6(6)^{2}

⇒ TSA of cube = 216dm^{2}

(c) Volume of cube = a^{3}

= (6)^{3}

⇒ Volume of cube = 216dm^{3}

**Question 3.**Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

2.5 m

**Answer:**Given:

side of cube (a) = 2.5m

(a) LSA of cube = 4a^{2}

= 4(2.5)^{2}

⇒ LSA of cube = 25m^{2}

(b) TSA of cube = 6a^{2}

= 6(2.5)^{2}

⇒ TSA of cube = 37.5m^{2}

(c) Volume of cube = a^{3}

= (2.5)^{3}

⇒ Volume of cube = 15.625m^{3}

**Question 4.**Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

24 cm

**Answer:**Given:

side of cube (a) = 24cm

(a) LSA of cube = 4a^{2}

= 4(24)^{2}

⇒ LSA of cube = 2304cm^{2}

(b) TSA of cube = 6a^{2}

= 6(24)^{2}

⇒ TSA of cube = 3456cm^{2}

(c) Volume of cube = a^{3}

= (24)^{3}

⇒ Volume of cube = 13824cm^{3}

**Question 5.**Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

31 cm

**Answer:**Given:

side of cube (a) = 31cm

(a) LSA of cube = 4a^{2}

= 4(31)^{2}

⇒ LSA of cube = 3844cm^{2}

(b) TSA of cube = 6a^{2}

= 6(31)^{2}

⇒ TSA of cube = 5766cm^{2}

(c) Volume of cube = a^{3}

= (31)^{3}

⇒ Volume of cube = 29791cm^{3}

**Question 6.**If the Lateral Surface Area of a cube is 900 cm^{2}, find the length of its side.

**Answer:**Given:

LSA = 900cm^{2}

LSA of cube = 4a^{2}

900 = 4(a)^{2}

⇒ (a)^{2} = 225

⇒ length of side of cube = 15cm

**Question 7.**If the Total Surface Area of a cube is 1014 cm^{2}, find the length of its side.

**Answer:**Given:

TSA = 1014cm^{2}

TSA of cube = 6a^{2}

1014 = 6(a)^{2}

⇒ (a)^{2} = 169

⇒ length of side of cube = 13cm

**Question 8.**The volume of the cube is 125 dm^{3}. Find its side.

**Answer:**Given:

Volume = 125dm^{3}

Volume of cube = a^{3}

125 = (a)^{3}

⇒ (a) = 5

⇒ length of side of cube = 5dm

**Question 9.**A container is in the shape of a cube of side 20 cm. How much sugar can it hold?

**Answer:**Given:

side of container (a) = 20cm

Volume of cube = a^{3}

= (20)^{3}

= 20 × 20 × 20

⇒ Volume of container = 8000cm^{3}

**Question 10.**A cubical tank can hold 64,000 litres of water. Find the length of the side of the tank.

**Answer:**Given:

Volume of tank = 64000 litres = 64m^{3}

Volume of cube = a^{3}

64000 = (a)^{3}

⇒ (a)^{3} = (4)^{3}

⇒ a = 4

⇒ length of side of tank = 4m

**Question 11.**Three metallic cubes of side 3 cm, 4 cm and 5 cm respectively are melted and are recast into a single cube. Find the total surface area of the new cube.

**Answer:**Given:

three cubes of side 3 cm, 4 cm and 5cm

Total volume of all cube = volume of new cube

(3)^{3} + (4)^{3} + (5)^{3} = (a)^{3}

27 + 64 + 125 = (a)^{3}

216 = (a)^{3}

⇒ a = 6

TSA of cube = 6a^{2}

= 6(6)^{2}

⇒ TSA of cube = 216cm^{2}

**Question 12.**How many cubes of side 3 cm are required to build a cube of side 15 cm?

**Answer:**Given:

side of small cube = 3cm

side of large cube = 15cm

volume of small cube = (3)^{3}

= 27

volume of large cube = (15)^{3}

= 3375

Number of required = volume of large cube/ volume of small cube

N = 3375/27

N = 125

Number of cubes required = 125

**Question 13.**Find the area of card board required to make an open cubical box of side 40 cm. Also find the volume of the box.

**Answer:**Given:

side of cube (a) = 40cm

Area of open cubical box = 5a^{2}

= 5(40)^{2}

⇒ area of cube = 8000cm^{2}

Volume of cube = a^{3}

= (40)^{3}

⇒ Volume of cube = 64000cm^{3}

**Question 14.**What is the total cost of oil in a cubical container of side 2 m if it is measured and sold using a cubical vessel of height 10 cm and the cost is ₹50 per measure.

**Answer:**Given:

side of container = 2m

side of vessel = 10cm = 0.1m

Volume of container = (2)^{3}

= 8000000cm^{3}

Volume of container = (10)^{3}

= 1000cm^{3}

Rate = 50 per 1000 m cube

So, cost =

Cost = Rs. 400000

**Question 15.**A container of side 3.5m is to be painted both inside and outside. Find the area to be painted and the total cost of painting it at the rate of ₹75 per square meter.

**Answer:**Given:

side of container (a) = 3.5cm

Area of cubical box to be painted both sides = 12a^{2}

= 12(3.5)^{2}

⇒ Area of cubical box to be painted both sides = 147m^{2}

total cost of painting it at the rate of Rs75 per square meter

then total cost of painting painted box = 147 × 75 = Rs11025

**Question 1.**

Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

5.6 cm

**Answer:**

Given:

side of cube (a) = 5.6cm

(a) LSA of cube = 4a^{2}

= 4(5.6)^{2}

⇒ LSA of cube = 125.44cm^{2}

(b) TSA of cube = 6a^{2}

= 6(5.6)^{2}

⇒ TSA of cube = 188.16cm^{2}

(c) Volume of cube = a^{3}

= (5.6)^{3}

⇒ Volume of cube = 175.62cm^{3}

**Question 2.**

Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

6 dm

**Answer:**

Given:

side of cube (a) = 6dm

(a) LSA of cube = 4a^{2}

= 4(6)^{2}

⇒ LSA of cube = 144dm^{2}

(b) TSA of cube = 6a^{2}

= 6(6)^{2}

⇒ TSA of cube = 216dm^{2}

(c) Volume of cube = a^{3}

= (6)^{3}

⇒ Volume of cube = 216dm^{3}

**Question 3.**

Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

2.5 m

**Answer:**

Given:

side of cube (a) = 2.5m

(a) LSA of cube = 4a^{2}

= 4(2.5)^{2}

⇒ LSA of cube = 25m^{2}

(b) TSA of cube = 6a^{2}

= 6(2.5)^{2}

⇒ TSA of cube = 37.5m^{2}

(c) Volume of cube = a^{3}

= (2.5)^{3}

⇒ Volume of cube = 15.625m^{3}

**Question 4.**

Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

24 cm

**Answer:**

Given:

side of cube (a) = 24cm

(a) LSA of cube = 4a^{2}

= 4(24)^{2}

⇒ LSA of cube = 2304cm^{2}

(b) TSA of cube = 6a^{2}

= 6(24)^{2}

⇒ TSA of cube = 3456cm^{2}

(c) Volume of cube = a^{3}

= (24)^{3}

⇒ Volume of cube = 13824cm^{3}

**Question 5.**

Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the cubes having their sides as

31 cm

**Answer:**

Given:

side of cube (a) = 31cm

(a) LSA of cube = 4a^{2}

= 4(31)^{2}

⇒ LSA of cube = 3844cm^{2}

(b) TSA of cube = 6a^{2}

= 6(31)^{2}

⇒ TSA of cube = 5766cm^{2}

(c) Volume of cube = a^{3}

= (31)^{3}

⇒ Volume of cube = 29791cm^{3}

**Question 6.**

If the Lateral Surface Area of a cube is 900 cm^{2}, find the length of its side.

**Answer:**

Given:

LSA = 900cm^{2}

LSA of cube = 4a^{2}

900 = 4(a)^{2}

⇒ (a)^{2} = 225

⇒ length of side of cube = 15cm

**Question 7.**

If the Total Surface Area of a cube is 1014 cm^{2}, find the length of its side.

**Answer:**

Given:

TSA = 1014cm^{2}

TSA of cube = 6a^{2}

1014 = 6(a)^{2}

⇒ (a)^{2} = 169

⇒ length of side of cube = 13cm

**Question 8.**

The volume of the cube is 125 dm^{3}. Find its side.

**Answer:**

Given:

Volume = 125dm^{3}

Volume of cube = a^{3}

125 = (a)^{3}

⇒ (a) = 5

⇒ length of side of cube = 5dm

**Question 9.**

A container is in the shape of a cube of side 20 cm. How much sugar can it hold?

**Answer:**

Given:

side of container (a) = 20cm

Volume of cube = a^{3}

= (20)^{3}

= 20 × 20 × 20

⇒ Volume of container = 8000cm^{3}

**Question 10.**

A cubical tank can hold 64,000 litres of water. Find the length of the side of the tank.

**Answer:**

Given:

Volume of tank = 64000 litres = 64m^{3}

Volume of cube = a^{3}

64000 = (a)^{3}

⇒ (a)^{3} = (4)^{3}

⇒ a = 4

⇒ length of side of tank = 4m

**Question 11.**

Three metallic cubes of side 3 cm, 4 cm and 5 cm respectively are melted and are recast into a single cube. Find the total surface area of the new cube.

**Answer:**

Given:

three cubes of side 3 cm, 4 cm and 5cm

Total volume of all cube = volume of new cube

(3)^{3} + (4)^{3} + (5)^{3} = (a)^{3}

27 + 64 + 125 = (a)^{3}

216 = (a)^{3}

⇒ a = 6

TSA of cube = 6a^{2}

= 6(6)^{2}

⇒ TSA of cube = 216cm^{2}

**Question 12.**

How many cubes of side 3 cm are required to build a cube of side 15 cm?

**Answer:**

Given:

side of small cube = 3cm

side of large cube = 15cm

volume of small cube = (3)^{3}

= 27

volume of large cube = (15)^{3}

= 3375

Number of required = volume of large cube/ volume of small cube

N = 3375/27

N = 125

Number of cubes required = 125

**Question 13.**

Find the area of card board required to make an open cubical box of side 40 cm. Also find the volume of the box.

**Answer:**

Given:

side of cube (a) = 40cm

Area of open cubical box = 5a^{2}

= 5(40)^{2}

⇒ area of cube = 8000cm^{2}

Volume of cube = a^{3}

= (40)^{3}

⇒ Volume of cube = 64000cm^{3}

**Question 14.**

What is the total cost of oil in a cubical container of side 2 m if it is measured and sold using a cubical vessel of height 10 cm and the cost is ₹50 per measure.

**Answer:**

Given:

side of container = 2m

side of vessel = 10cm = 0.1m

Volume of container = (2)^{3}

= 8000000cm^{3}

Volume of container = (10)^{3}

= 1000cm^{3}

Rate = 50 per 1000 m cube

So, cost =

Cost = Rs. 400000

**Question 15.**

A container of side 3.5m is to be painted both inside and outside. Find the area to be painted and the total cost of painting it at the rate of ₹75 per square meter.

**Answer:**

Given:

side of container (a) = 3.5cm

Area of cubical box to be painted both sides = 12a^{2}

= 12(3.5)^{2}

⇒ Area of cubical box to be painted both sides = 147m^{2}

total cost of painting it at the rate of Rs75 per square meter

then total cost of painting painted box = 147 × 75 = Rs11025

###### Exercise 4.3

**Question 1.**Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

5 cm, 2 cm , 11cm

**Answer:**Given:

l = 5cm, b = 2cm and h = 11cm

(a) LSA of cuboid = 2(lh+hb)

= 2(5(11) + 2(11))

= 2(55+22)

= 2(77) = 154cm^{2}

⇒ LSA of cuboid = 154cm^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(5(11) + 2(11) + 5(2))

= 2(55+22+10)

⇒ TSA of cuboid = 174cm^{2}

(c) Volume of cuboid = l× b× h

= 5× 2× 11

⇒ Volume of cuboid = 110cm^{3}

**Question 2.**Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

15 dm, 10 dm, 8 dm

**Answer:**Given:

l = 15dm, b = 10dm and h = 8dm

(a) LSA of cuboid = 2(lh+hb)

= 2(15(8) + 8(10))

= 2(120+80)

= 2(200) = 400dm^{2}

⇒ LSA of cuboid = 400dm^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(15(8) + 8(10) + 15(10))

= 2(120+80+150)

⇒ TSA of cuboid = 700dm^{2}

(c) Volume of cuboid = l× b× h

= 15× 10× 8

⇒ Volume of cuboid = 1200dm^{3}

**Question 3.**Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

2 m, 3 m, 7 m

**Answer:**Given:

l = 2m, b = 3m and h = 7m

(a) LSA of cuboid = 2(lh+hb)

= 2(2(7) + 7(3))

= 2(14+21)

= 2(35) = 70m^{2}

⇒ LSA of cuboid = 66m^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(2(7) + 7(3) + 2(3))

= 2(14+21+6)

⇒ TSA of cuboid = 82m^{2}

(c) Volume of cuboid = l× b× h

= 2× 3× 7

⇒ Volume of cuboid = 42m^{3}

**Question 4.**Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

20 m, 12 m, 8 m

**Answer:**Given:

l = 20m, b = 12m and h = 8m

(a) LSA of cuboid = 2(lh+hb)

= 2(20(8) + 8(12))

= 2(160+96)

= 2(256) = 512m^{2}

⇒ LSA of cuboid = 512m^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(20(8) + 8(12) + 20(12))

= 2(160+96+240)

⇒ TSA of cuboid = 992m^{2}

(c) Volume of cuboid = l× b× h

= 20× 12× 8

⇒ Volume of cuboid = 1920m^{3}

**Question 5.**Find the height of the cuboid whose length, breadth and volume are 35 cm, 15 cm and 14175 cm^{3} respectively.

**Answer:**Given:

l = 35cm, b = 15cm and volume = 14175 cm^{3}

Volume of cuboid = l× b× h

14175 = 35× 15× h

⇒ h = 14175/ 525

⇒ h = 27

⇒ Height of cuboid = 27cm

**Question 6.**Two cubes each of volume 64 cm^{3} are joined to form a cuboid. Find the L.S.A and T.S.A of the resulting solid.

**Answer:**Given:

volume of given cube = 64cm^{3}

Side = a cm

Volume = (a)^{3}

64 = (a)^{3}

a = 4cm

two cubes are joined together.

Length of resultant cube = 4+4 = 8cm

Breadth = 4cm

Height = 4cm

(a) LSA of cuboid = 2(lh+hb)

= 2(8(4) + 4(4))

= 2(32+16)

= 2(48) = 96cm^{2}

⇒ LSA of cuboid = 66m^{2}

(b)TSA of cuboid = 2(lh+hb+lb)

= 2(8(4) + 4(4) + 8(4))

= 2(32+16+32)

⇒ TSA of cuboid = 2(80) = 160m^{2}

**Question 7.**Raju planned to stitch a cover for his two speaker boxes whose length, breadth andheight are 35 cm, 30 cm and 55 cm respectively. Find the cost of the cloth he has to buy if its costs ₹75 per sq.m.

**Answer:**Given:

l = 35cm, b = 30cm and h = 55cm

Total area to cover two speaker =2{ 2(lh + bh) + lb)}

= 2{2(35(55) + 30(55)) + 35(30)}

=2{ 2(1925 + 1650 )+ 1050}

=2{ 2(210) + 1050 }

= 2{7150 + 1050} = 16400 cm^{2} = 1.64m^{2}

⇒ area = 1.64m^{2}

Cost of 1m = Rs 75

Cost of 1.64 m^{2} = 1.64× 75

= Rs 123

**Question 8.**Mohan wanted to paint the walls and ceiling of a hall. The dimensions of the hall is 20m x15m x6m. Find the area of surface to be painted and the cost of painting it at ₹78 per sq. m.

**Answer:**Given:

l = 20m, b = 15m and h = 6m

area of walls and ceiling to be renovated = 2(lh + bh) + lb)

= 2(20(6) + 15(6)) + 20(15)

= 2(120 + 90 )+ 300

= 2(210) + 300

= 420 + 300 = 720 m^{2}

⇒ area = 720m^{2}

Cost of 1m = Rs 78

Cost of 720 m^{2} = 720× 48

= Rs 56160

**Question 9.**How many hollow blocks of size 30cm x 15cm x20cm are needed to construct a wall 60m in length, 0.3m in breadth and 2m in height.

**Answer:**Given:

sides of small cuboid = 30cm x 15cm x20cm and sides of large cuboid = 60m x 0.3m x 2m

volume of small cube = 30cm x 15cm x20cm

= 9000cm^{3} = 0.009m^{3}

volume of large cube = 60m x 0.3m x 2m

= 36m

Number of hollow blocks required = volume of large cuboid/ volume of small cuboid

N = 36/0.009

N = 4000

Number of hollow blocks required = 4000

**Question 10.**Find the cost of renovating the walls and the floor of a hall that measures 10m x 45m x 6m if the cost is ₹48 per square meter.

**Answer:**Given:

l = 10m, b = 45m and h = 6m

area of cuboid to be renovated = 2(lh + bh)+ lb

= 2(10(6) + 45(6)) + 10(45)

= 2(60 + 270) +450

= 660 + 450 = 1110

⇒ area of cuboid to be renovated zz= 1110m^{3}

Cost of 1m = Rs 48

Cost of 1110 m^{2} = 1110× 48

= Rs 53280

**Question 1.**

Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

5 cm, 2 cm , 11cm

**Answer:**

Given:

l = 5cm, b = 2cm and h = 11cm

(a) LSA of cuboid = 2(lh+hb)

= 2(5(11) + 2(11))

= 2(55+22)

= 2(77) = 154cm^{2}

⇒ LSA of cuboid = 154cm^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(5(11) + 2(11) + 5(2))

= 2(55+22+10)

⇒ TSA of cuboid = 174cm^{2}

(c) Volume of cuboid = l× b× h

= 5× 2× 11

⇒ Volume of cuboid = 110cm^{3}

**Question 2.**

Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

15 dm, 10 dm, 8 dm

**Answer:**

Given:

l = 15dm, b = 10dm and h = 8dm

(a) LSA of cuboid = 2(lh+hb)

= 2(15(8) + 8(10))

= 2(120+80)

= 2(200) = 400dm^{2}

⇒ LSA of cuboid = 400dm^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(15(8) + 8(10) + 15(10))

= 2(120+80+150)

⇒ TSA of cuboid = 700dm^{2}

(c) Volume of cuboid = l× b× h

= 15× 10× 8

⇒ Volume of cuboid = 1200dm^{3}

**Question 3.**

Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

2 m, 3 m, 7 m

**Answer:**

Given:

l = 2m, b = 3m and h = 7m

(a) LSA of cuboid = 2(lh+hb)

= 2(2(7) + 7(3))

= 2(14+21)

= 2(35) = 70m^{2}

⇒ LSA of cuboid = 66m^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(2(7) + 7(3) + 2(3))

= 2(14+21+6)

⇒ TSA of cuboid = 82m^{2}

(c) Volume of cuboid = l× b× h

= 2× 3× 7

⇒ Volume of cuboid = 42m^{3}

**Question 4.**

Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height

respectively as

20 m, 12 m, 8 m

**Answer:**

Given:

l = 20m, b = 12m and h = 8m

(a) LSA of cuboid = 2(lh+hb)

= 2(20(8) + 8(12))

= 2(160+96)

= 2(256) = 512m^{2}

⇒ LSA of cuboid = 512m^{2}

(b) TSA of cuboid = 2(lh+hb+lb)

= 2(20(8) + 8(12) + 20(12))

= 2(160+96+240)

⇒ TSA of cuboid = 992m^{2}

(c) Volume of cuboid = l× b× h

= 20× 12× 8

⇒ Volume of cuboid = 1920m^{3}

**Question 5.**

Find the height of the cuboid whose length, breadth and volume are 35 cm, 15 cm and 14175 cm^{3} respectively.

**Answer:**

Given:

l = 35cm, b = 15cm and volume = 14175 cm^{3}

Volume of cuboid = l× b× h

14175 = 35× 15× h

⇒ h = 14175/ 525

⇒ h = 27

⇒ Height of cuboid = 27cm

**Question 6.**

Two cubes each of volume 64 cm^{3} are joined to form a cuboid. Find the L.S.A and T.S.A of the resulting solid.

**Answer:**

Given:

volume of given cube = 64cm^{3}

Side = a cm

Volume = (a)^{3}

64 = (a)^{3}

a = 4cm

two cubes are joined together.

Length of resultant cube = 4+4 = 8cm

Breadth = 4cm

Height = 4cm

(a) LSA of cuboid = 2(lh+hb)

= 2(8(4) + 4(4))

= 2(32+16)

= 2(48) = 96cm^{2}

⇒ LSA of cuboid = 66m^{2}

(b)TSA of cuboid = 2(lh+hb+lb)

= 2(8(4) + 4(4) + 8(4))

= 2(32+16+32)

⇒ TSA of cuboid = 2(80) = 160m^{2}

**Question 7.**

Raju planned to stitch a cover for his two speaker boxes whose length, breadth andheight are 35 cm, 30 cm and 55 cm respectively. Find the cost of the cloth he has to buy if its costs ₹75 per sq.m.

**Answer:**

Given:

l = 35cm, b = 30cm and h = 55cm

Total area to cover two speaker =2{ 2(lh + bh) + lb)}

= 2{2(35(55) + 30(55)) + 35(30)}

=2{ 2(1925 + 1650 )+ 1050}

=2{ 2(210) + 1050 }

= 2{7150 + 1050} = 16400 cm^{2} = 1.64m^{2}

⇒ area = 1.64m^{2}

Cost of 1m = Rs 75

Cost of 1.64 m^{2} = 1.64× 75

= Rs 123

**Question 8.**

Mohan wanted to paint the walls and ceiling of a hall. The dimensions of the hall is 20m x15m x6m. Find the area of surface to be painted and the cost of painting it at ₹78 per sq. m.

**Answer:**

Given:

l = 20m, b = 15m and h = 6m

area of walls and ceiling to be renovated = 2(lh + bh) + lb)

= 2(20(6) + 15(6)) + 20(15)

= 2(120 + 90 )+ 300

= 2(210) + 300

= 420 + 300 = 720 m^{2}

⇒ area = 720m^{2}

Cost of 1m = Rs 78

Cost of 720 m^{2} = 720× 48

= Rs 56160

**Question 9.**

How many hollow blocks of size 30cm x 15cm x20cm are needed to construct a wall 60m in length, 0.3m in breadth and 2m in height.

**Answer:**

Given:

sides of small cuboid = 30cm x 15cm x20cm and sides of large cuboid = 60m x 0.3m x 2m

volume of small cube = 30cm x 15cm x20cm

= 9000cm^{3} = 0.009m^{3}

volume of large cube = 60m x 0.3m x 2m

= 36m

Number of hollow blocks required = volume of large cuboid/ volume of small cuboid

N = 36/0.009

N = 4000

Number of hollow blocks required = 4000

**Question 10.**

Find the cost of renovating the walls and the floor of a hall that measures 10m x 45m x 6m if the cost is ₹48 per square meter.

**Answer:**

Given:

l = 10m, b = 45m and h = 6m

area of cuboid to be renovated = 2(lh + bh)+ lb

= 2(10(6) + 45(6)) + 10(45)

= 2(60 + 270) +450

= 660 + 450 = 1110

⇒ area of cuboid to be renovated zz= 1110m^{3}

Cost of 1m = Rs 48

Cost of 1110 m^{2} = 1110× 48

= Rs 53280

###### Exercise 4.4

**Question 1.**The length of the arc of a sector having central angle 90° and radius 7 cm is

A. 22 cm

B. 44 cm

C. 11 cm

D. 33 cm

**Answer:**Given:

length of arc = ?

radius = 7cm

Arc length =

⇒ Arc length = 11cm

Option (C) is correct.

**Question 2.**If the radius and arc length of a sector are 17 cm and 27 cm respectively, then the perimeter is

A. 16 cm

B. 61 cm

C. 32 cm

D. 80 cm

**Answer:**Given:

Perimeter = ?

length of arc = 27cm

radius = 17cm

perimeter of sector = arc length + r + r

= arc length + 2r

= 2(17) + 27

= 34 + 27 = 61cm

Option (B) is correct.

**Question 3.**If the angle subtended by the arc of a sector at the center is 90°, then the area of the sector in square units is

A. 2πr^{2}

B. 4πr^{2}

C.

D.

**Answer:**Given:

radius = r

central angle = 90°

Area of sector =

Option (C) is correct.

**Question 4.**Area of a sector having radius 12 cm and arc length 21 cm is

A. 126 cm^{2}

B. 252 cm^{2}

C. 33 cm^{2}

D. 45 cm^{2}.

**Answer:**Given:

radius = 12cm

arc length = 21cm

Area of sector =

⇒ area of sector = 126cm

Option (A) is correct.

**Question 5.**The area of a sector with radius 4 cm and central angle 60° is

A.

B.

C.

D.

**Answer:**Given:

radius = 4cm

central angle = 60°

Area of sector =

⇒ central angle = 8π /3 cm^{2}

Option (C) is correct.

**Question 6.**If the area and arc length of the sector of a circle are 60 cm^{2} and 20 cm respectively, then the diameter of the circle is

A. 6 cm

B. 12 cm

C. 24 cm

D. 36 cm

**Answer:**Given:

area of sector = 60cm^{2}

arc length = 20cm

Area of sector =

⇒ Radius of circle = 6cm

Diameter of circle = 12cm

Option (B) is correct.

**Question 7.**The perimeter of a sector of a circle is 37cm. If its radius is 7cm, then its arc length is

A. 23 cm

B. 5.29 cm

C. 32 cm

D. 259 cm

**Answer:**Given:

Perimeter = 37cm

radius = 7cm

perimeter of sector = arc length + r + r

= arc length + 2r

37 = 2(7) + a

a = 37 - 14 = 23cm

Option (a) is correct.

**Question 8.**A solid having six equal square faces is called a

A. cube

B. cuboid

C. square

D. rectangle

**Answer:**A solid having six equal square faces is Cube.

Option (a) is correct.

**Question 9.**The quantity of space occupied by a body is its

A. area

B. length

C. volume

D. T.S.A

**Answer:**Quantity of space occupied by a body is its volume.

Option (c) is correct.

**Question 10.**The LSA of a cube of side 1dm is

A. 16 dm^{2}

B. 4 dm^{2}

C. 2 dm^{2}

D. 1 dm^{2}

**Answer:**Given:

side of cube (a) = 1dm

LSA of cube = 4a^{2}

= 4(1)^{2}

= 4(1)

⇒ LSA of cube = 4dm^{2}

Option (B) is correct.

**Question 1.**

The length of the arc of a sector having central angle 90° and radius 7 cm is

A. 22 cm

B. 44 cm

C. 11 cm

D. 33 cm

**Answer:**

Given:

length of arc = ?

radius = 7cm

Arc length =

⇒ Arc length = 11cm

Option (C) is correct.

**Question 2.**

If the radius and arc length of a sector are 17 cm and 27 cm respectively, then the perimeter is

A. 16 cm

B. 61 cm

C. 32 cm

D. 80 cm

**Answer:**

Given:

Perimeter = ?

length of arc = 27cm

radius = 17cm

perimeter of sector = arc length + r + r

= arc length + 2r

= 2(17) + 27

= 34 + 27 = 61cm

Option (B) is correct.

**Question 3.**

If the angle subtended by the arc of a sector at the center is 90°, then the area of the sector in square units is

A. 2πr^{2}

B. 4πr^{2}

C.

D.

**Answer:**

Given:

radius = r

central angle = 90°

Area of sector =

Option (C) is correct.

**Question 4.**

Area of a sector having radius 12 cm and arc length 21 cm is

A. 126 cm^{2}

B. 252 cm^{2}

C. 33 cm^{2}

D. 45 cm^{2}.

**Answer:**

Given:

radius = 12cm

arc length = 21cm

Area of sector =

⇒ area of sector = 126cm

Option (A) is correct.

**Question 5.**

The area of a sector with radius 4 cm and central angle 60° is

A.

B.

C.

D.

**Answer:**

Given:

radius = 4cm

central angle = 60°

Area of sector =

⇒ central angle = 8π /3 cm^{2}

Option (C) is correct.

**Question 6.**

If the area and arc length of the sector of a circle are 60 cm^{2} and 20 cm respectively, then the diameter of the circle is

A. 6 cm

B. 12 cm

C. 24 cm

D. 36 cm

**Answer:**

Given:

area of sector = 60cm^{2}

arc length = 20cm

Area of sector =

⇒ Radius of circle = 6cm

Diameter of circle = 12cm

Option (B) is correct.

**Question 7.**

The perimeter of a sector of a circle is 37cm. If its radius is 7cm, then its arc length is

A. 23 cm

B. 5.29 cm

C. 32 cm

D. 259 cm

**Answer:**

Given:

Perimeter = 37cm

radius = 7cm

perimeter of sector = arc length + r + r

= arc length + 2r

37 = 2(7) + a

a = 37 - 14 = 23cm

Option (a) is correct.

**Question 8.**

A solid having six equal square faces is called a

A. cube

B. cuboid

C. square

D. rectangle

**Answer:**

A solid having six equal square faces is Cube.

Option (a) is correct.

**Question 9.**

The quantity of space occupied by a body is its

A. area

B. length

C. volume

D. T.S.A

**Answer:**

Quantity of space occupied by a body is its volume.

Option (c) is correct.

**Question 10.**

The LSA of a cube of side 1dm is

A. 16 dm^{2}

B. 4 dm^{2}

C. 2 dm^{2}

D. 1 dm^{2}

**Answer:**

Given:

side of cube (a) = 1dm

LSA of cube = 4a^{2}

= 4(1)^{2}

= 4(1)

⇒ LSA of cube = 4dm^{2}

Option (B) is correct.