Volume & Capacity

Mensuration is the branch of mathematics that deals with measurement of lengths, areas and volumes of geometric figures. It covers two-dimensional figures (areas) and three-dimensional figures (volumes and capacities). In this chapter we focus on volume and capacity, their formulae, units and worked examples involving common solids such as cubes, cuboids, cylinders and cones.

What is Volume?

Volume is the measure of the amount of space occupied by a three-dimensional (3-D) object. A solid must have three dimensions - length, breadth (or width) and height (or thickness) - to have volume. For a hollow object, the volume of the interior space is its capacity (discussed later).

For example, a cuboid is a solid bounded by six rectangles. If its length is l, breadth is b and height is h, then the space occupied by the cuboid (its volume) is given by the product of its three dimensions.

Volume of a cuboid:

\(V = l \times b \times h\)

What is Capacity?

Capacity is the amount of liquid or gas a hollow container can hold. Capacity measures the interior volume of a container when it is filled. Thus capacity is volume of the hollow part of an object and is used especially for liquids and gases.

Examples of objects for which we commonly use capacity are a cylinder (cup), a cone (cup-shaped vessels), a hollow hemisphere, tanks and bottles.

Note on units: Volume is expressed in cubic units such as cubic centimetre (cm³), cubic metre (m³), etc. Capacity is commonly expressed in litres (L) and millilitres (mL). The relation between the units is:

• 1 cm³ = 1 mL
• 1000 mL = 1 L
• 1 m = 100 cm, so 1 m³ = 100 × 100 × 100 = 1 000 000 cm³

Mensuration of Some Enclosed Figures

Cube

A cube is a solid with six equal square faces. If the edge (side) of the cube is a, then each face has area a × a and the cube has six such faces.

Total surface area of a cube:

\( \text{T.S.A.} = 6a^2 \)

Volume of a cube:

\( V = a \times a \times a = a^3 \)

Example: Find the width of the cube whose volume is 625cm³.

Solution: 

Given volume is 625 cm³.

\( a^3 = 625 \)

\( a = \sqrt[3]{625} \)

\( a \approx 8.549 \text{ cm (approximately)} \)

Cuboid

A cuboid is a solid with six rectangular faces. Its three dimensions are length (l), breadth or width (b) and height (h). The total surface area is the sum of the areas of all six faces.

Total surface area of a cuboid:

\( \text{T.S.A.} = 2(lb + bh + hl) \)

Volume of a cuboid:

\( V = l \times b \times h \)

Example: Find the height of a cuboid if its volume is 625 cm³ and its base area is 25 cm².

Solution: 

Volume of cuboid \(V = l \times b \times h\).

Base area \(= l \times b = 25 \text{ cm}^2\).

\( 625 = (l \times b) \times h = 25 \times h \)

\( h = \dfrac{625}{25} = 25 \text{ cm} \)

Cylinder (Right Circular Cylinder)

A right circular cylinder has a circular base of radius r and height h. Common examples include pipes, cans and pillars. The volume of the cylinder is the area of the base multiplied by the height.

Volume of a cylinder:

\( V = \pi r^2 h \)

Example: A rectangular sheet of paper having length 11 cm and width 4 cm is being rolled to form a cylinder of height 4 cm. What is the volume of the cylinder

Solution: 

The width of the sheet becomes the height of the cylinder, so \(h = 4\) cm.

The length of the sheet becomes the circumference of the base of the cylinder, so \(2\pi r = 11\) cm.

\( r = \dfrac{11}{2\pi} = \dfrac{11}{2 \times \tfrac{22}{7}} = \dfrac{7}{4} \text{ cm} \)

\( V = \pi r^2 h \)

\( V = \dfrac{22}{7} \times \left(\dfrac{7}{4}\right)^2 \times 4 \)

\( V = \dfrac{22}{7} \times \dfrac{49}{16} \times 4 = 38.5 \text{ cm}^3 \)

Hence, the volume of the cylinder is 38.5 cm³.

Cone

A right circular cone has a circular base of radius r and a perpendicular height h from the base to the apex. The volume of a cone is one third of the volume of a cylinder with the same base and height.

Volume of a cone:

\( V = \dfrac{1}{3} \pi r^2 h \)

Sample Problems on Volume and Capacity

Problem 1: Find the capacity, in litres, of a cubic tank with length, width, height given as 1 m, 1 m and 1 m respectively.

Solution: 

The tank is a cube of side 1 m, so its volume is \(1 \text{ m}^3\).

Convert to cubic centimetres: \(1 \text{ m} = 100 \text{ cm}\).

\( 1 \text{ m}^3 = 100 \times 100 \times 100 = 1\,000\,000 \text{ cm}^3 \)

Since \(1 \text{ cm}^3 = 1 \text{ mL}\), the capacity is \(1\,000\,000 \text{ mL}\).

Convert to litres: \(1\,000 \text{ mL} = 1 \text{ L}\).

\( \text{Capacity} = \dfrac{1\,000\,000}{1\,000} = 1\,000 \text{ L} \)

The capacity of the tank is 1000 litres.

Problem 2: A tank measures 2 m, 1 m, 2 m. Find the capacity of the tank.

Solution:

The tank is a cuboid with dimensions \(2 \text{ m} \times 1 \text{ m} \times 2 \text{ m}\).

\( \text{Volume} = 2 \times 1 \times 2 = 4 \text{ m}^3 \)

Convert to cubic centimetres: \(1 \text{ m} = 100 \text{ cm}\).

\( 4 \text{ m}^3 = 4 \times (100)^3 = 4 \times 1\,000\,000 = 4\,000\,000 \text{ cm}^3 \)

Since \(1 \text{ cm}^3 = 1 \text{ mL}\), this is \(4\,000\,000 \text{ mL}\).

Convert to litres: \( \dfrac{4\,000\,000}{1\,000} = 4\,000 \text{ L} \)

The capacity of the tank is 4000 litres.

Problem 3: A rectangular sheet of paper is being rolled and converted to a cylinder of radius 10 cm. What is the volume of the cylinder? Width of rectangle is 7 cm.

Solution:

The width of the rectangle becomes the height of the cylinder, so \(h = 7\) cm.

Radius of the base \(r = 10\) cm.

\( V = \pi r^2 h \)

\( V = \dfrac{22}{7} \times 10^2 \times 7 \)

\( V = \dfrac{22}{7} \times 100 \times 7 = 22 \times 100 = 2200 \text{ cm}^3 \)

The volume of the cylinder is 2200 cm³.

Problem 4: What is the volume of the cone if the radius is 9 cm and height is 14 cm

Solution:

\( V = \dfrac{1}{3} \pi r^2 h \)

Substitute \(r = 9\) cm and \(h = 14\) cm.

\( V = \dfrac{1}{3} \times \dfrac{22}{7} \times 9^2 \times 14 \)

Simplify step by step:

\( 9^2 = 81 \)

\( \dfrac{22}{7} \times 81 \times 14 = 22 \times 81 \times 2 = 22 \times 162 \)

\( \dfrac{1}{3} \times 22 \times 162 = 22 \times 54 = 1188 \)

\( V = 1188 \text{ cm}^3 \)

The volume of the cone is 1188 cm³.

Summary and Useful Formulae

• Cube: Volume = \(a^3\); T.S.A. = \(6a^2\).
• Cuboid: Volume = \(lbh\); T.S.A. = \(2(lb + bh + hl)\).
• Cylinder: Volume = \(\pi r^2 h\); curved surface area = \(2\pi r h\); T.S.A. = \(2\pi r(r + h)\).
• Cone: Volume = \(\dfrac{1}{3}\pi r^2 h\); curved surface area = \(\pi r l\) where l is slant height; T.S.A. = \(\pi r (r + l)\).
• Unit conversions: \(1 \text{ cm}^3 = 1 \text{ mL}\), \(1000 \text{ mL} = 1 \text{ L}\), \(1 \text{ m}^3 = 1\,000\,000 \text{ cm}^3\).