Surface Areas and Volumes of Solid Shapes

# Surface Areas and Volumes of Solid Shapes

## 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 km3 = 1,00,00,00,000 m3
1 hm3 = 10,00,000 m3
1 dam3 = 1,000 m3
1 dm3 = 0.001 m3
1 cm3 = 0.000001 m3
1 mm3 = 0.000000001 m3

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 cm3,
1 L = 1000 cm3
1 kL = 1 m3 = 1000000 cm3

## 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 = 4a2, a is the side of the cube.
Surface area of a cube = 6a2, a is the side of the cube.
Volume of a cube = a3, 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€r2h, 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 2h, 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€r2, r is the radius of the sphere.
Volume of a sphere = 4/3 Ï€r3, r is the radius of the sphere.

## Surface Area and Volume of a Hemisphere

Surface area of a hemisphere = 3Ï€r2, r is the radius of the hemisphere.
Volume of a hemisphere = 2/3 Ï€r3, 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, d2 = l2 + b2 + h2
(25)2 = l2 + (16)2 + (12)2
l2 = (25)2 – [(16)2 + (12)2] = 625 – 400 = 225
l = 15 cm
Volume of a cuboid = l × b × h = 15 × 16 × 12 cm3 = 2880 cm3
The length of the cuboid is 15 cm and its volume is 2880 cm3.

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 = 4a2 = 4 × 92 = 4 × 81 = 324 m2
Total surface area = 6a2 = 6 × 92 = 6 × 81 = 486 m2

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€r2h
= 22/7 × 7 × 7 × 4 cm3
= 22 × 7 × 4 cm3 = 616 cm3
Thus, the volume of the cylinder is 616 cm3.

Example 4: The total surface area of a cuboid is 13950 cm2 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 2x, 3x and 5x, respectively.
According to the question, total surface area = 2(lb + bh + lh) = 13950
13950 = 2(2x × 3x + 3x × 5x + 2x × 5x)
13950 = 2(6x2 + 15x2 + 10x2)
13950 = 2 × 31x2
62x2 = 13950
x2 = 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 cm2 = 11250 cm2
The lateral surface area of the cuboid is 11250 cm2.

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 cm2

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 m2.

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 m2
= 2 × 22 × 0.05 × 4 m2
= 8.8 m2
So, curved surface area of 20 pillars = 20 × 8.8 m2
= 176 m2

Cost of painting = Rs 10 × 176 = Rs 1760