Volume

Video lesson3 worked examples

Theory

You must be able to calculate the volume of standard solids:

Sphere ( $V = \frac{4}{3}\pi r^3$ )
Cone ( $V = \frac{1}{3}\pi r^2 h$ )
Pyramid ( $V = \frac{1}{3}Ah$ )
Cylinder ( $V = \pi r^2 h$ )
Prism ( $V = Ah$ )

Complex problems will require calculating the volume of composite solids (shapes joined together) or simple fractional parts of solids (e.g., hemispheres).

The Golden Rule: choose the right formula, and check whether you are given the radius or the diameter. For a composite solid, add the parts (or subtract, if one is removed). If the volume is given and a length is unknown, substitute and rearrange.

⚠️ Common Examiner Traps

Diameter given, not radius: if told the diameter, halve it first — a very common lost mark.
The $\frac{1}{3}$ for cones and pyramids: forgetting it triples the answer.
Units: volume is in cubic units (cm³, m³); make sure all lengths are in the same unit before calculating.
Rounding: if asked for a number of significant figures, keep full accuracy until the end.

Worked examples

Example 1

Volume of a Cone

A cone has a base radius of 6 cm and a height of 14 cm. Calculate its volume.

Step 1: Substitute into the formula: $V = \frac{1}{3} \times \pi \times 6^2 \times 14$ .

Step 2: Calculate: $\frac{1}{3} \times \pi \times 36 \times 14 = 12 \times 14 \times \pi$ .

Answer: $168\pi$ cm³ (or approx. 527.79 cm³).

Example 2

Composite Solid (Hemisphere + Cylinder)

A silo is formed by a cylinder of radius 4 m and height 10 m, topped by a hemisphere of the same radius. Find the total volume.

Step 1: Cylinder volume: $V_1 = \pi \times 4^2 \times 10 = 160\pi$ .

Step 2: Hemisphere volume: $V_2 = \frac{1}{2} \times (\frac{4}{3}\pi \times 4^3) = \frac{1}{2} \times \frac{256\pi}{3} = \frac{128\pi}{3}$ .

Step 3: Add them together: $160\pi + 42.67\pi$ .

Answer: 202.67 $\pi$ m³ (or approx. 636.70 m³).

Example 3

Working Backwards (finding a length)

A pyramid has a square base of side 6 cm and a volume of 96 cm³. Calculate its height.

Step 1: The base area is $6 \times 6 = 36$ cm². Substitute the known values into $V = \frac{1}{3}Ah$ :

96 = \frac{1}{3} \times 36 \times h

Step 2: Simplify the right-hand side: $96 = 12h$ .

Step 3: Divide to find $h$ : $h = 8$ .

Answer: the height is 8 cm.

Ready to practise?

12 Volume questions with answers and video solutions