Volume of a cuboid and nets
How much water does a fish tank hold, and how much card do you need to make a box for a present? Both questions are about a cuboid: one asks about the space inside it, the other about the surface around it. By the end of this page you will be able to answer both, and you will find out how many litres a tank 80 cm long, 40 cm wide and 50 cm high holds.
What is a cuboid?
A cuboid is a solid shape with six rectangular faces. A shoebox, a brick, a cereal packet and a fish tank are all cuboids. A cube is a special cuboid in which all the edges are the same length.
Three words describe the parts of a cuboid:
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a face is a flat surface (6 of them);
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an edge is where two faces meet (12 of them);
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a vertex (plural vertices) is a corner where edges meet (8 of them).
The faces come in three pairs of identical rectangles: front and back, top and bottom, left and right. This is the key to everything that follows.
How many faces, edges and vertices does a cuboid have?
How many faces, edges and vertices does a cuboid have?
Volume of a cuboid
Volume is the amount of space inside a solid. We measure it by counting unit cubes. A cube with edges of 1 cm has a volume of 1 cm³ (one cubic centimetre).
Take a cuboid 5 cm long, 4 cm wide and 3 cm high. One layer on the bottom holds $5 \times 4 = 20$ unit cubes. There are 3 layers, so there are $20 \times 3 = 60$ cubes in all.
Why it works: the base area tells you how many cubes fit in one layer, and the height tells you how many layers there are.
For a cube with edge $a$: $V = a \times a \times a = a^3$.
Mass is often given in kilograms, but heavy loads are measured in tonnes: $1000$ kg $= 1$ tonne. To change kilograms to tonnes, divide by $1000$.
A cuboid measures $5$ cm by $4$ cm by $3$ cm. Find its volume.
A cuboid measures $5$ cm by $4$ cm by $3$ cm. Find its volume.
A fish tank in a Galway aquarium shop is $80$ cm long, $40$ cm wide and $50$ cm high. a) Find its volume in cm³. b) Given that $1000$ cm³ $= 1$ litre, how many litres of…
A fish tank in a Galway aquarium shop is $80$ cm long, $40$ cm wide and $50$ cm high.
- a) Find its volume in cm³.
- b) Given that $1000$ cm³ $= 1$ litre, how many litres of water does it hold when full?
A skip on a building site in Dublin is a cuboid $3$ m long, $1.5$ m wide and $1.2$ m deep (open top). It is filled with rubble to $\frac{2}{3}$ of its depth. a) Find the…
A skip on a building site in Dublin is a cuboid $3$ m long, $1.5$ m wide and $1.2$ m deep (open top). It is filled with rubble to $\frac{2}{3}$ of its depth.
- a) Find the volume of rubble in m³.
- b) Each m³ of rubble has a mass of $1,600$ kg. Find the total mass in tonnes.
Nets and surface area
A net is a flat shape that folds up to make a solid. Imagine cutting along some edges of a box and opening it out flat.
The net of a cuboid has six rectangles. Matching colours show the three pairs of identical faces.
The total surface area is the area of the whole net, that is, the sum of all six faces. For a cuboid with length $l$, width $w$ and height $h$:
A cube has edges of length $6$ cm. a) Find its volume. b) Find its total surface area.
A cube has edges of length $6$ cm.
- a) Find its volume.
- b) Find its total surface area.
The net of a closed box is made of six rectangles. The box is $10$ cm long, $6$ cm wide and $4$ cm high. a) List the dimensions of each pair of identical rectangles in…
The net of a closed box is made of six rectangles. The box is $10$ cm long, $6$ cm wide and $4$ cm high.
- a) List the dimensions of each pair of identical rectangles in the net.
- b) Find the total area of the net.
A cuboid has volume $240$ cm³, length $10$ cm and width $6$ cm. a) Find its height. b) Find its total surface area.
A cuboid has volume $240$ cm³, length $10$ cm and width $6$ cm.
- a) Find its height.
- b) Find its total surface area.
Using letters and scaling a cube
Sometimes the sides are given as letters. Write the volume as an expression, then use it to form an equation.
What happens if you double every edge? Take a cube of edge $a$. Its surface area is $6a^2$ and its volume is $a^3$. If the edge becomes $2a$:
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surface area: $6(2a)^2 = 6 \times 4a^2 = 4 \times 6a^2$, so it is multiplied by 4;
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volume: $(2a)^3 = 8a^3$, so it is multiplied by 8.
In general, if every edge is multiplied by $k$, the areas are multiplied by $k^2$ and the volume by $k^3$.
A cuboid has a square base of side $x$ cm and height $2x$ cm. Its volume is $432$ cm³. a) Write an expression for the volume in terms of $x$. b) Find $x$. c) Find the…
A cuboid has a square base of side $x$ cm and height $2x$ cm. Its volume is $432$ cm³.
- a) Write an expression for the volume in terms of $x$.
- b) Find $x$.
- c) Find the total surface area.
A cube has edge $a$ cm. Each edge is doubled. a) Show that the surface area is multiplied by $4$. b) Show that the volume is multiplied by $8$. c) A cube of edge $3$ cm…
A cube has edge $a$ cm. Each edge is doubled.
- a) Show that the surface area is multiplied by $4$.
- b) Show that the volume is multiplied by $8$.
- c) A cube of edge $3$ cm has its edge tripled. By what factor does the volume change, and what is the new volume?
Summary
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A cuboid has 6 faces (three identical pairs), 12 edges and 8 vertices. A cube is a cuboid with all edges equal.
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Volume = length × width × height = base area × height. Use the same unit for all lengths. Answers are in cm³ or m³; 1000 cm³ = 1 litre.
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A net is the cuboid opened out flat. Surface area = area of the net = $2lw + 2lh + 2wh$; for a cube, $6a^2$.
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To find a missing side, put the numbers into the volume formula and solve the equation.
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If every edge is multiplied by $k$, surface area is multiplied by $k^2$ and volume by $k^3$.