Volume & capacity

Year 7 core

By the end of this topic you should be able to:

distinguish volume (space occupied by a solid) from capacity (how much a container can hold),
convert between mm^3, cm^3, m^3 and mL, L, kL,
calculate the volume of a rectangular prism (cuboid),
calculate the volume of a triangular prism,
apply these skills to tanks, boxes, and swimming pools.

1. Volume and capacity

Volume is the amount of three-dimensional space a solid occupies. Units are cubed: mm^3, cm^3, m^3.

Capacity is how much liquid a container holds. Units are mL, L, kL.

The two are linked: $\;1$ cm^3 of water is exactly $1$ mL.

Conversions

Volume units

1 \text{ cm}^{3} = 1000 \text{ mm}^{3}, \quad 1 \text{ m}^{3} = 1\,000\,000 \text{ cm}^{3}.

Capacity units

1 \text{ L} = 1000 \text{ mL}, \quad 1 \text{ kL} = 1000 \text{ L}.

Linking volume and capacity

1 \text{ cm}^{3} = 1 \text{ mL}, \qquad 1 \text{ m}^{3} = 1000 \text{ L} = 1 \text{ kL}.

2. Volume of a right prism

A right prism has two identical, parallel end-faces (the bases) joined by rectangular side-faces. For any right prism, the volume is the area of its base multiplied by its length.

Volume of any right prism

V \;=\; \text{area of base} \times \text{length}.

3. Rectangular prism (cuboid)

A rectangular prism has a rectangular base.

A rectangular prism (cuboid) has length L, width W and height H.

Rectangular prism

V = L \times W \times H.

Worked example 1 Volume of a box

A box measures $8$ cm long, $5$ cm wide, $4$ cm high. Find its volume.

V = 8 \times 5 \times 4 = 160 \text{ cm}^{3}.

How many millilitres of water could this box hold?

160 \text{ cm}^{3} = 160 \text{ mL}.

4. Triangular prism

A triangular prism has a triangular base. Calculate the area of the triangle, then multiply by the length of the prism.

Triangular prism

V = \tfrac{1}{2} \times b \times h \times \ell,

where $b$ and $h$ are the base and perpendicular height of the triangle, and $\ell$ is the length of the prism.

Worked example 2 Volume of a tent

A ridge tent has a triangular cross-section of base $2.4$ m and height $1.8$ m, and is $3$ m long. Find the volume.

Area of triangle: $\tfrac{1}{2} \times 2.4 \times 1.8 = 2.16$ m^2.
Volume: $2.16 \times 3 = 6.48$ m^3.

5. Finding a missing dimension

Worked example 3 Finding height from a capacity

A rectangular tank has a base $2$ m by $1.5$ m and holds $9000$ L of water. What is its depth?

Convert capacity: $9000$ L $= 9$ m^3.
$V = L \times W \times H$ , so $9 = 2 \times 1.5 \times H$ .
$9 = 3H$ , so $H = 3$ m.

Practice: Year 7 core

Fluency

Unit conversions

Convert $2500$ mL to litres.
Convert $4.6$ L to mL.
Convert $7500$ cm^3 to mL.
Convert $3$ m^3 to litres.
Convert $4500$ mm^3 to cm^3.
A $2$ L bottle holds how many cm^3?

Fluency

Rectangular prism

Find the volume of a $6 \times 5 \times 4$ cm cuboid.
Find the volume of a cube with edge $7$ cm.
Find the volume of a $12 \times 8 \times 3$ m room.
A fish tank is $40 \times 25 \times 20$ cm. Find the volume in cm^3 and the capacity in L.
A cube has volume $125$ cm^3. Find the edge length.
A rectangular tank has base $80$ cm by $60$ cm and height $50$ cm. Find the capacity in litres.

Fluency

Triangular prism

A triangular prism has a triangular base of base $6$ cm and height $4$ cm, and length $10$ cm. Find the volume.
A tent has a triangular cross-section of base $2$ m and height $1.5$ m, and is $2.5$ m long. Find the volume.
A wedge-shaped doorstop has a right-triangular base with legs $4$ cm and $6$ cm, and is $8$ cm wide. Find the volume.
A triangular prism has volume $120$ cm^3. Its length is $10$ cm. What is the area of the triangular base?

Reasoning

Explain and reason

Ben writes the volume of a $4 \times 4 \times 4$ cube as $4^2 = 16$ cm^3. What mistake has Ben made?
Explain in your own words why $1$ cm^3 $= 1$ mL.
Two rectangular tanks have the same capacity. Must they have the same surface area? Give a reason or a counter-example.
Without calculating, decide which has the greater volume: a cube of side $6$ cm, or a rectangular prism of $5 \times 6 \times 7$ cm. Explain briefly.
A rectangular prism and a triangular prism both have length $10$ cm. The rectangular prism has a $6$ cm by $4$ cm base. What base area would the triangular prism need so that they have the same volume?

Problem solving

Real-world problems

A water tank is $1.2$ m by $0.8$ m by $1.5$ m deep. How many litres when full?
A swimming pool is $15$ m long, $8$ m wide, and has a uniform depth of $1.5$ m. How many kilolitres? At $2.50/kL, what is the cost to fill?
A shoebox is $33$ cm $\times$ $22$ cm $\times$ $15$ cm. Find the volume in cm^3 and in litres (to $2$ dp).
A small aquarium holds $36$ L and has base $60$ cm by $30$ cm. What is the water height?
A $2.5$ L carton is poured into glasses that hold $250$ mL each. How many full glasses?
A chocolate bar is a triangular prism with equilateral cross-section (side $3$ cm, height $\approx 2.6$ cm) and length $12$ cm. Find its volume (to the nearest cm^3).