G16, G17 - Volume and surface area of prisms | AQA GCSE Mathematics

Volume measures the space inside a solid; surface area measures its exposed outside. In G16 you will calculate volumes of cuboids, other right prisms and cylinders. In G17 you will use familiar face areas to find their surface areas, including containers with an open end.

Cuboid volume and capacity

A cylindrical pot has radius $3\,\mathrm{cm}$ and height $8\,\mathrm{cm}$ . How much less material is needed if its lid is left off? Its capacity stays the same, but one exposed face disappears. By the end, you will calculate exactly which area is saved.

You already know

Rectangle area is length times width; circle area is $\pi r^2$ . Surface area uses square units; volume uses cubic units. $1000\,\mathrm{cm}^3=1\,\mathrm{L}$ .

Picture a cuboid filled with $1\,\mathrm{cm}$ cubes. A base $8\,\mathrm{cm}$ by $5\,\mathrm{cm}$ holds $40$ cubes in one layer. Three layers hold $120$ cubes. Multiplying by the height counts the layers, rather than another face.

Cuboid volume

For a cuboid with length $l$ , width $w$ and perpendicular height $h$ , in matching units,

V=lwh

The volume uses cubic units. For a cube of side $s$ , this becomes $V=s^3$ .

These forms are not printed separately on the formulae sheet: derive them from the prism formula, which is on the 2027 formulae sheet.

For a container's capacity, use its internal dimensions. Outside dimensions would include the thickness of its walls.

Worked example 1: counting layers in a box

Question. A cuboid has dimensions $8\,\mathrm{cm}$ , $5\,\mathrm{cm}$ and $3\,\mathrm{cm}$ . Work out its volume. Non-calculator.

Plan. Find the rectangular base area, then multiply by height.

\begin{aligned} 8\times5&=40 &&\text{base area in square centimetres}\\ 40\times3&=120 &&\text{three centimetres of layers} \end{aligned}

Why use all three lengths? A solid has depth as well as the two dimensions of a face.

Answer. $120\,\mathrm{cm}^3$ .

Check. Three layers of $40$ unit cubes contain $120$ cubes.

Change the unknown from volume to height. The layer area still comes first, but now divide the volume by it.

Worked example 2: an unknown internal height

Question. A rectangular container has internal base $8\,\mathrm{cm}$ by $5\,\mathrm{cm}$ and capacity $180\,\mathrm{mL}$ . Work out its internal height. Non-calculator.

Plan. Use $180\,\mathrm{mL}=180\,\mathrm{cm}^3$ , then divide by the base area.

\begin{aligned} V&=8\times5\times h &&\text{cuboid volume}\\ 180&=40h &&\text{known capacity in cubic centimetres}\\ h&=180\div40=4.5 &&\text{height in centimetres} \end{aligned}

Why divide by $40$ ? Each centimetre of height holds $40\,\mathrm{cm}^3$ .

Answer. $4.5\,\mathrm{cm}$ .

Check. $8\times5\times4.5=180\,\mathrm{cm}^3$ , the supplied capacity.

The calculation can have fractional dimensions even though unit cubes helped explain the formula.

Watch out

Wrong: $8\times5+3$ for a cuboid's volume. This adds a length to an area. Right: multiply the base area by the perpendicular height.

Treat the next capacity as a volume before solving for height.

Right-prism cross-sections

A right prism keeps the same cross-section all the way along its length, so its volume is the area of one slice multiplied by the length.

Right prism

A solid with equal, parallel polygonal end faces and a constant cross-section, with its joining edges perpendicular to the end faces. A cuboid and a triangular right prism are examples. A pyramid is not: its cross-sections shrink towards the top.

The cross-section may be a triangle, trapezium or composite polygon. Its height belongs inside the area calculation. The prism's length is a different measurement: the distance between its two ends.

[DIAGRAM: asset_name: Triangular right-prism cross-section and length; asset_slug: aqa8300_gd_prism_cross_section; caption: Find the triangular cross-section area before multiplying by the prism length; description: A triangular right prism has a highlighted blue end face with perpendicular base 8 cm and height 6 cm. The distance between the end faces is 15 cm. Text gives cross-section area 24 square centimetres and volume 360 cubic centimetres. Not drawn accurately.]
Find the triangular cross-section area before multiplying by the prism length

Prism volume

For a right prism with constant cross-section area $A$ and length $l$ ,

V=Al

The length is perpendicular to the cross-section. Use matching units.

On the 2027 formulae sheet.

Writing the cross-section area on its own line makes the two stages much easier to follow.

Worked example 3: a triangular prism

Question. A right prism has triangular cross-section with base $8\,\mathrm{cm}$ and perpendicular height $6\,\mathrm{cm}$ . Its length is $15\,\mathrm{cm}$ . Work out its volume. Non-calculator.

Plan. Find the area of the triangular end, then multiply by $15$ .

\begin{aligned} A&=\frac12\times8\times6=24 &&\text{cross-section area}\\ V&=24\times15 &&\text{area times prism length}\\ &=360 &&\text{volume in cubic centimetres} \end{aligned}

Why halve only once? The triangular cross-section is already accounted for before multiplying by length.

Answer. $360\,\mathrm{cm}^3$ .

Check. The prism fills half an $8$ by $6$ by $15$ cuboid, whose volume is $720\,\mathrm{cm}^3$ .

Change the cross-section to a trapezium. The prism length and cross-section height stay the same; only the end-face area calculation changes.

Worked example 4: a trapezoidal channel

Question. A right prism has a trapezium cross-section with parallel sides $6\,\mathrm{cm}$ and $10\,\mathrm{cm}$ , separated by $6\,\mathrm{cm}$ . Its length is $15\,\mathrm{cm}$ . Work out its volume. Non-calculator.

Plan. Use the trapezium area formula for the end, then the prism volume formula.

\begin{aligned} A&=\frac12(6+10)\times6 &&\text{area of one end}\\ &=48 &&\text{square centimetres}\\ V&=48\times15=720 &&\text{multiply by prism length} \end{aligned}

Why is $6\,\mathrm{cm}$ not the prism length? Here it is the separation of the parallel sides within the end face. The ends are $15\,\mathrm{cm}$ apart.

Answer. $720\,\mathrm{cm}^3$ .

Check. The trapezium's mean width is $8\,\mathrm{cm}$ , giving $8\times6\times15=720$ .

For a composite cross-section, add or subtract its component areas first. Once its area is known, the volume step is unchanged.

Watch out

Wrong: use the perimeter of the end face in $V=Al$ . Right: $A$ is an area, measured in square units. Area times length gives cubic units.

Separate the end-face height from the prism length in this question.

Cylinder volume

A right cylinder uses the same constant-cross-section idea, with a circular end instead of a polygon.

Every slice perpendicular to its axis is the same circle. Its cross-section area is $\pi r^2$ , so multiplying by the perpendicular height gives its volume. The curved side does not need to be measured to find the space inside.

Cylinder volume

For a right cylinder of radius $r$ and perpendicular height $h$ ,

V=\pi r^2h

Use matching length units. The answer is in cubic units.

This form is not printed separately: derive it from the circle area and prism volume formulae on the 2027 formulae sheet.

Read the given circle measurement first. A diameter must be halved before it enters the squared part of the formula.

Worked example 5: cylinder volume from radius

Question. A cylinder has radius $3\,\mathrm{cm}$ and height $8\,\mathrm{cm}$ . Work out its volume in terms of $\pi$ . Non-calculator.

Plan. Find the circular cross-section area, then multiply by height.

\begin{aligned} A&=\pi\times3^2=9\pi &&\text{cross-section area}\\ V&=9\pi\times8 &&\text{multiply by height}\\ &=72\pi &&\text{simplify the coefficient} \end{aligned}

Why square $3$ but not $8$ ? The radius appears twice in the circle's area; height adds the third dimension once.

Answer. $72\pi\,\mathrm{cm}^3$ .

Check. Since $\pi$ is just over $3$ , the volume is just over $216\,\mathrm{cm}^3$ .

Change only the supplied circle measurement: give the same cylinder's diameter instead of its radius.

Worked example 6: cylinder volume from diameter

Question. A cylinder has diameter $6\,\mathrm{cm}$ and height $8\,\mathrm{cm}$ . Work out its volume to $3$ significant figures. Calculator allowed.

Plan. Halve the diameter, then use $\pi r^2h$ without rounding intermediate values.

\begin{aligned} r&=6\div2=3 &&\text{radius in centimetres}\\ V&=\pi\times3^2\times8 &&\text{cylinder volume}\\ &=226.1946711\ldots &&\text{calculator value} \end{aligned}

Why halve before squaring? Squaring the diameter would make the cross-section area four times too large.

Answer. $226\,\mathrm{cm}^3$ to $3$ significant figures.

Check. The cylinder is the same size as in Worked example 5, and $72\pi=226.1946711\ldots$ .

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Algebra

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Geometry and measures

Probability

Statistics

G16, G17 - Volume and surface area of prisms

Cuboid volume and capacity

Right-prism cross-sections

Cylinder volume

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