**Volume **

Volume is
the amount of space occupied by a solid shape. The volume of a solid shape is
expressed in cubic units. The volume of liquids is expressed in liters and
milliliters.

**Difference between Volume
and Capacity **

Volume is
the space that a certain solid object occupies whereas capacity tells about the
quantity of a liquid or gas a container or object can hold or accommodate.

Volume
is measured mostly in cubic centimeters (cu cm) or cubic meters (cu m) whereas
capacity is measured in liters or milliliters.

**Units of Volume and Capacity**

The fundamental
unit of volume is cubic meter. There are other units of volume also. We can convert
the units of volume according to the following relations.

1 km

^{3}= 1,00,00,00,000 m^{3}
1 hm

^{3}= 10,00,000 m^{3}
1 dam

^{3}= 1,000 m^{3}
1 dm m

^{3}= 0.001^{3}
1 cm

^{3}= 0.000001 m^{3}
1 mm

^{3}= 0.000000001 m^{3}
The capacity is measured in terms of liters (L). The smaller
unit of capacity is milliliter (mL) and the larger unit of capacity is kiloliter (kL).

1 kiloliter = 1000 liters

1 milliliter =
0.001 liter or 1 liter = 1000 milliliters

Also,

1 mL = 1 cm

^{3},
1 L = 1000 cm

1 kL = 1 m^{3}^{3}= 1000000 cm

^{3}

**Formulas for Surface Areas and Volumes**

The surface area and volume of various solid shapes are
given below:

**Surface
Area and Volume of a Cuboid**

Lateral surface area of a cuboid = 2(l + b)h, where l,
b, and h are the length, breadth and height of a cuboid.

Surface area of a cuboid = 2(lb + bh + hl), where l, b,
and h are the length, breadth and height of a cuboid.

Volume of a cuboid = l * b * h, where l, b and h are the length,
breadth and height of a cuboid.

**Surface Area and Volume of a ****Cube**** **

Lateral surface area of a cube = 4a

^{2}, a is the side of the cube.
Surface area of a cube = 6a

^{2}, a is the side of the cube.
Volume of a cube = a

^{3}, a is the side of the cube.**Surface
Area and Volume of a Cylinder**** **

Lateral or curved surface area of a cylinder = 2Ï€rh, r
is the radius of circular base and h is the height of the cylinder.

Total surface area of a cylinder = 2Ï€r(r + h), r is the
radius of circular base and h is the height of the cylinder.

Volume of a cylinder = Ï€r

^{2}h, r is the radius of circular base and h is the height of the cylinder.**Surface Area and Volume of a ****Cone**** **

Lateral or curved surface area of a cone = Ï€rL, r
is the radius of the circular base, L is the slant height of the cone.

Surface area of a cone = Ï€r(L+r), where r is the radius
of the circular base, L is the slant height of the cone.

Volume of a cone = 1/3 Ï€r

^{2}h, where r is the radius of the circular base, h is the perpendicular height of the cone.**Surface Area and Volume of a ****Sphere **** **

Surface area of a sphere = 4Ï€r

^{2}, r is the radius of the sphere.
Volume of a sphere = 4/3 Ï€r

^{3}, r is the radius of the sphere.**Surface Area and Volume of a ****Hemisphere**** **

Surface area of a hemisphere = 3Ï€r

^{2}, r is the radius of the hemisphere.
Volume of a hemisphere = 2/3 Ï€r

^{3}, r is the radius of the hemisphere.**Solved Examples on ****Surface Area and Volume**

**Example 1:**The diagonal of a cuboid is 25 cm, breadth is 16 cm and height is 12 cm. Find its length and volume.

**Solution:**Given,

*d*= 25 cm

We know that,

*d*^{2}=*l*^{2}+*b*^{2}+*h*^{2}
(25)

^{2}=*l**+ (16)*^{2}^{2}+ (12)^{2}*l*

*= (25)*

^{2}^{2}– [(16)

^{2}+ (12)

^{2}] = 625 – 400 = 225

*l*= 15 cm

Volume of a cuboid =

*l × b × h =*15 × 16 × 12 cm^{3}= 2880 cm^{3}
The length of the cuboid is 15 cm and its volume is
2880 cm

^{3}.**Example 2:**Find the lateral surface area and the total surface area of a cube whose edge is 9 m.

**Solution:**Given:

*a*= 9 m = edge

Lateral surface area = 4

*a*^{2}= 4 × 9^{2}= 4 × 81 = 324 m^{2}
Total surface area = 6

*a*^{2}= 6 × 9^{2}= 6 × 81 = 486 m^{2}**Example 3:**Find the volume of a cylinder whose base radius is 7 cm and height is 4 cm.

(Take Ï€ = 22/7)

**Solution:**Radius of the base (

*r*) = 7 cm

Height of the
cylinder (

*h*) = 4 cm
Volume of the
cylinder = Ï€

*r*^{2}*h*
= 22/7 × 7 × 7 × 4 cm

^{3}
= 22 × 7 × 4 cm

^{3}= 616 cm^{3}
Thus, the volume of the
cylinder is 616 cm

^{3}.**Example 4:**The total surface area of a cuboid is 13950 cm

^{2}and its length, breadth and height are given in the ratio 2 : 3 : 5. Find the lateral surface area of the cuboid.

**Solution:**Let the length, breadth and height of the cuboid be 2

*x*, 3

*x*and 5

*x*, respectively.

According to the question, total surface area = 2(

*lb + bh + lh*) = 13950
13950 = 2(2

*x*× 3*x*+ 3*x*× 5*x*+ 2*x*× 5*x*)
13950 = 2(6

*x*^{2}+ 15*x*^{2}+ 10*x*^{2})
13950 = 2 × 31

*x*^{2}
62

*x*^{2}= 13950*x*

^{2}= 225

*x*= 15 cm

Thus, length = 2 × 15, breadth = 3 × 15, height = 5 ×
15

*l*= 30 cm,

*b*= 45 cm,

*h*= 75 cm

Lateral surface area = 2(

*l + b*) ×*h*= 2(30 + 45) × 75 cm^{2}= 11250 cm^{2}
The lateral surface area of the cuboid is 11250 cm

^{2}.**Example 5:**Find the total surface area of a hollow cylinder whose external and internal radii are 4 cm and 3 cm, respectively and height is 14 cm.

**Solution:**Given: Radii R = 4 cm,

*r*= 3 cm,

*h*= 14 cm

Total surface area of the
hollow cylinder

= 2Ï€ (R
+

*r*) (*h*+ R –*r*)
= 2 × 22/7 × (4 + 3) ×
(14 + 4 – 3)

= 2 × 22/7 × 7 × 15 =
660 cm

^{2}**Example 6:**In a building, there are 20 identical pillars that are cylinders. The radius and height of each pillar are 35 cm and 4 m, respectively. Find the total cost of painting the curved surface area of all the pillars at the rate of Rs 10 per m

^{2}.

**Solution:**

**Radius of the cylindrical pillar (**

*r*) = 35 cm = 0.35 m

Height (

*h*) = 4 m
Curved surface
area of a cylinder = 2Ï€

*rh*
Curved surface
area of a pillar = 2 × 22/7 × 0.35 × 4 m

^{2}
= 2 × 22 × 0.05 × 4 m

^{2}
= 8.8 m

^{2}
So, curved
surface area of 20 pillars = 20 × 8.8 m

^{2}
= 176 m

^{2}
Cost of painting = Rs 10
× 176 = Rs
1760