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Volume and capacity

Find the volume of prisms, cylinders, pyramids, cones and spheres; convert between volume and capacity; apply the trapezoidal rule to estimate the volume of irregular solids.

Worked examples

Volume of a cylinder

Straightforward

Problem

A cylindrical water tank has a radius of 22 m and a height of 55 m. Find the volume of the tank. Give your answer to 2 decimal places.

Volume of a cone and converting to capacity

Moderate

Problem

A conical container has a radius of 66 cm and a perpendicular height of 1414 cm. Find the volume of the container, then state the capacity in millilitres. Give the volume to 2 decimal places.

Volume using the trapezoidal rule

Challenging

Problem

A section of a river has an irregular cross-section. The width at the front is 88 m and the width at the rear is 1212 m, with a perpendicular distance of 33 m between the two measurements. The section of river being measured is 2525 m long. Use the trapezoidal rule A≈h2(df+dl)A \approx \dfrac{h}{2}(d_f + d_l) to estimate the volume of water in this section.

Practise

Q1·Straightforward
A rectangular storage crate has a length of 88 m, a width of 55 m, and a height of 33 m. Find the volume of the crate.
Q2·Straightforward
A triangular prism has a triangular cross-section with a base of 66 m and a perpendicular height of 44 m. The prism is 99 m long. Find the volume of the prism.
Q3·Straightforward
A cylindrical pipe has a radius of 44 cm and a length of 1010 cm. Find the volume of the pipe. Give your answer to 2 decimal places.
Q4·Straightforward
A cylindrical water tank has a radius of 33 m and a height of 77 m. Find the volume of the tank. Give your answer to 2 decimal places.
Q5·Moderate
A square pyramid has a base of side length 66 m and a perpendicular height of 44 m. Find the volume of the pyramid.
Q6·Moderate
A conical funnel has a radius of 55 cm and a perpendicular height of 1212 cm. Find the volume of the cone. Give your answer to 2 decimal places.
Q7·Moderate
A spherical ball has a radius of 33 cm. Find the volume of the ball. Give your answer to 2 decimal places.
Q8·Moderate
A rectangular fish tank has internal dimensions 5050 cm long, 3030 cm wide, and 2020 cm tall. Find the capacity of the tank in litres. (Use 11 cm3^3 =1= 1 mL.)
Q9·Challenging
A concrete retaining wall has an irregular cross-section. The depth at the front is df=3d_f = 3 m and the depth at the rear is dl=8d_l = 8 m, with a perpendicular distance of h=4h = 4 m between the two measurements. The wall is 3030 m long. Use the trapezoidal rule A≈h2(df+dl)A \approx \dfrac{h}{2}(d_f + d_l) to estimate the volume of concrete in the wall.
Q10·Challenging
A section of an irrigation channel has an irregular cross-section. The width at the top is df=1.6d_f = 1.6 m and the width at the base is dl=0.8d_l = 0.8 m, with a perpendicular depth of h=2.5h = 2.5 m. The section is 1212 m long. Use the trapezoidal rule A≈h2(df+dl)A \approx \dfrac{h}{2}(d_f + d_l) to estimate the volume of water the section holds.
Q11·Challenging
An earth embankment has an irregular cross-section. The width at the top is df=6d_f = 6 m and the width at the base is dl=2d_l = 2 m, with a perpendicular height of h=5h = 5 m. The embankment is 4545 m long. Use the trapezoidal rule A≈h2(df+dl)A \approx \dfrac{h}{2}(d_f + d_l) to estimate the volume of earth in the embankment.
Q12·Challenging
A garden bed has an irregular cross-section. The width at the front edge is df=4.2d_f = 4.2 m and the width at the rear edge is dl=6.8d_l = 6.8 m, with a perpendicular distance of h=3h = 3 m between the two edges. The garden bed is 2020 m long. Use the trapezoidal rule A≈h2(df+dl)A \approx \dfrac{h}{2}(d_f + d_l) to estimate the volume of soil needed to fill the garden bed.