G12, G16, G17Geometry & MeasuresFoundation & HigherGrades 3–8

Volume & Surface Area

Volume is the space inside a 3D shape, measured in cubic units; surface area is the total area of all its faces, measured in square units. One idea covers most solids — a prism's volume is its cross-section area times its length — and a few more formulas handle pyramids, cones and spheres.

50 min Video by Maths Genie AQA GCSE Maths

The big picture

A prism has the same cross-section all the way through, so its volume is that area "stacked up" along the length. Cuboids and cylinders are just prisms with rectangular and circular cross-sections.

Surface area is 2D thinking applied to a 3D shape: unfold it into a net and add the areas of the pieces. For a cylinder, the curved surface unrolls into a rectangle whose length is the circumference.

Surface AreaCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Calculate the volume of cuboids, prisms and cylinders
Calculate the surface area of cuboids, prisms and cylinders
Convert between volume units, including cm³, m³ and litres
Use formulas for pyramids, cones and spheres (Higher)

Prisms and cylinders

Volume of a prism $=$ area of cross-section $\times$ length. For a cuboid, $V = lwh$ . For a cylinder, the cross-section is a circle: $V = \pi r^2 h$ .

Identify the cross-section first — it is the face that is constant through the shape, which may not be the one it is resting on.

A triangular prism: the shaded cross-section

A

runs the whole length

l

Example 1A triangular prism has a cross-section with base 6 cm and height 4 cm, and length 15 cm. Find its volume.

1Cross-section:

\frac{1}{2} \times 6 \times 4 = 12

cm².

2Volume

= 12 \times 15 = 180

Answer

180

cm³

Your turnSame method, new numbers — try it before moving on.

A cylinder has radius 5 cm and height 12 cm. Find its volume, to the nearest cm³.

Your answer

Show worked solution

Solution

1 $V = \pi \times 5^2 \times 12 = 300\pi$ .
2 $= 942.48\ldots$ , so $942$ cm³.

Answer: $942$ cm³

Example 2A cuboid tank has a base 40 cm by 25 cm and holds 30 litres of water. How deep is the water?

130 litres

= 30{,}000

cm³.

2Base area

= 1000

cm².

3Depth

= 30{,}000 \div 1000 = 30

cm.

Answer

30

Your turnSame method, new numbers — try it before moving on.

A cuboid has volume 360 cm³, length 12 cm and width 5 cm. Find its height, in cm.

Your answer

Show worked solution

Solution

1 $12 \times 5 \times h = 360$ .
2 $60h = 360$ , so $h = 6$ cm.

Answer: $6$ cm

Surface area

Find the area of every face and add them. A cuboid has three pairs of identical rectangles: $SA = 2(lw + lh + wh)$ .

A cylinder has two circular ends and a curved surface that unrolls into a rectangle of width $2\pi r$ and height $h$ : $SA = 2\pi r^2 + 2\pi r h$ .

Example 1Find the surface area of a cuboid 8 cm by 5 cm by 3 cm.

1Pairs of faces:

8 \times 5 = 40

8 \times 3 = 24

5 \times 3 = 15

SA = 2(40 + 24 + 15) = 158

Answer

158

cm²

Your turnSame method, new numbers — try it before moving on.

Find the total surface area of a cube with edges of 7 cm, in cm².

Your answer

Show worked solution

Solution

1Each face: $7 \times 7 = 49$ cm².
2Six faces: $6 \times 49 = 294$ cm².

Answer: $294$ cm²

Example 2Find the total surface area of a closed cylinder with radius 3 cm and height 10 cm, in terms of

\pi

1Two ends:

2 \times \pi \times 9 = 18\pi

2Curved:

2\pi \times 3 \times 10 = 60\pi

Answer

78\pi

cm²

Tip — An open box or a pipe has fewer faces — read whether the shape is closed before adding the ends.

Volume units

$1$ cm³ $= 10 \times 10 \times 10 = 1000$ mm³ and $1$ m³ $= 100^3 = 1{,}000{,}000$ cm³. Cube the length factor.

Capacity: $1$ litre $= 1000$ cm³, and $1$ m³ $= 1000$ litres.

Example 1A pond holds 2.4 m³ of water. How many litres is this?

1

m³

= 1000

litres.

Answer

2400

litres

Your turnSame method, new numbers — try it before moving on.

Convert 0.35 m³ to cm³.

Your answer

Show worked solution

Solution

1 $0.35 \times 1{,}000{,}000 = 350{,}000$ cm³.

Answer: $350{,}000$ cm³

Pyramids, cones and spheres (Higher)

Pyramid and cone: $V = \frac{1}{3} \times$ base area $\times$ perpendicular height, so a cone is $V = \frac{1}{3}\pi r^2 h$ . Sphere: $V = \frac{4}{3}\pi r^3$ .

Curved surface area of a cone $= \pi r l$ , where $l$ is the slant height (use Pythagoras if needed). Surface area of a sphere $= 4\pi r^2$ . These formulas are given in the exam when needed — but you must know how to use them.

Example 1A cone has radius 6 cm and perpendicular height 8 cm. Find its volume and its total surface area, in terms of

\pi

1Volume

= \frac{1}{3} \times \pi \times 36 \times 8 = 96\pi

cm³.

2Slant height

l = \sqrt{6^2 + 8^2} = 10

3Curved surface

= \pi \times 6 \times 10 = 60\pi

; base

= 36\pi

. Total

= 96\pi

cm².

Answer

Volume

96\pi

cm³; surface area

96\pi

cm²

Your turnSame method, new numbers — try it before moving on.

A sphere has radius 3 cm. Find its volume in terms of $\pi$ .

Your answer

Type π as pi, e.g. 12pi.

Show worked solution

Solution

1 $V = \frac{4}{3} \times \pi \times 27 = 36\pi$ cm³.

Answer: $36\pi$ cm³

Example 2A solid is a hemisphere of radius 4 cm on top of a cylinder of radius 4 cm and height 10 cm. Find its volume to 3 s.f.

1Cylinder:

\pi \times 16 \times 10 = 160\pi

2Hemisphere:

\frac{1}{2} \times \frac{4}{3}\pi \times 64 = \frac{128}{3}\pi

3Total

= \frac{608}{3}\pi = 636.7\ldots

Answer

637

cm³

Where the one-third comes from

A cone fits inside a cylinder with the same base and height, and holds exactly one third of the cylinder's volume — you can verify it by pouring sand. The $\frac{1}{3}$ in the pyramid and cone formulas is that fact.

Think like an examiner

State the cross-section area as a separate step — it is usually the first method mark.
Check units match before calculating (all cm or all m) and give cubic units for volume.
For composite solids, write the volume of each part with a label before adding.

Volume formulas

V_{\text{prism}} = A \times l, \quad V_{\text{cyl}} = \pi r^2 h

Prisms and cylinders.

SA_{\text{cyl}} = 2\pi r^2 + 2\pi rh

Closed cylinder.

V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h, \quad V_{\text{sphere}} = \tfrac{4}{3}\pi r^3

Higher solids.

Watch out for these

Using the slant height of a cone in

\frac{1}{3}\pi r^2 h

Volume uses the perpendicular height; the slant height is for the curved surface.

1

m³

= 100

cm³.

1

m³

= 1{,}000{,}000

cm³.

Cylinder surface area without the two circular ends.

A closed cylinder needs

2\pi r^2

as well.

Volume of a sphere as

\frac{4}{3}\pi r^2

Volume is cubic:

\frac{4}{3}\pi r^3

Stretch yourself

A cylindrical glass of radius 4 cm is filled with water to a depth of 12 cm. A solid metal sphere of radius 3 cm is dropped in and sinks completely. By how much does the water level rise? Give your answer to 2 decimal places.

Hint — The rise in water volume equals the volume of the sphere. Divide by the base area of the glass.

Reveal solution

SolutionSphere volume

= \frac{4}{3}\pi \times 27 = 36\pi

cm³. Base area of glass

= 16\pi

cm². Rise

= \frac{36\pi}{16\pi} = 2.25

cm. (The water, now 14.25 cm deep, still covers the 6 cm sphere.)

\blacksquare

Questions students ask

Are the cone and sphere formulas given in the exam?

AQA gives the formulas for the volume and surface area of cones and spheres, and the volume of a pyramid, inside questions where they are needed. Prism, cylinder and cuboid formulas are not given.

What is the difference between capacity and volume?

Capacity is how much a container holds, usually in litres or millilitres.

1

= 1

cm³.

How do I find the surface area of a triangular prism?

Add the two triangular ends and the three rectangular sides. Sketching the net makes sure no face is missed.

Key takeaways

Prism volume = cross-section area × length.
Cylinder: $V = \pi r^2 h$ and $SA = 2\pi r^2 + 2\pi rh$ .
Surface area adds the areas of all faces.
Cube the length factor for volume units; $1$ litre $= 1000$ cm³.
Cone $\frac{1}{3}\pi r^2 h$ ; sphere $\frac{4}{3}\pi r^3$ .

Exam-style questions

Practise it the way it is examined

3 original questions · 10 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 33 marks

A cuboid box measures 30 cm by 20 cm by 15 cm.

(a)Work out the volume of the box.[1]
Show mark scheme (a)
- B1 $9000$ cm³
Answer: $9000$ cm³
(b)How many cubes of side 5 cm fit exactly inside the box?[2]
Show mark scheme (b)
- M1 $6 \times 4 \times 3$ or $9000 \div 125$
- A1 $72$
Answer: $72$

Question 2Grade 53 marks

A cylindrical tin has diameter 8 cm and height 11 cm. A label covers the curved surface only.

Work out the area of the label. Give your answer to 1 decimal place.[3]
Show mark scheme
- M1Uses radius 4 or circumference $8\pi$
- M1 $\pi \times 8 \times 11$
- A1 $276.5$ cm² (276.46…)
Answer: $276.5$ cm²

Question 3Grade 84 marks

A solid metal cone has radius 6 cm and height 10 cm. It is melted down and recast into spheres of radius 1.5 cm. $\left[V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h, \ V_{\text{sphere}} = \tfrac{4}{3}\pi r^3\right]$

How many complete spheres can be made?[4]
Show mark scheme
- M1Cone volume $= \frac{1}{3}\pi \times 36 \times 10 = 120\pi$
- M1Sphere volume $= \frac{4}{3}\pi \times 3.375 = 4.5\pi$
- M1 $120\pi \div 4.5\pi = 26.66\ldots$
- A1 $26$ complete spheres
Answer: $26$

Examiner note

Rounding 26.67 up to 27 loses the final mark — there is not enough metal for a 27th sphere.

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Volume of a cuboid 4 cm × 5 cm × 6 cm?
- 120 cm³
- 15 cm³
- 148 cm³
- 60 cm³
Show answer and explanation
Answer: 120 cm³
$4 \times 5 \times 6$ .
A prism has cross-section 18 cm² and length 10 cm. Volume?
- 180 cm³
- 28 cm³
- 1800 cm³
- 90 cm³
Show answer and explanation
Answer: 180 cm³
Area × length.
1 litre is equal to…
- 1000 cm³
- 100 cm³
- 1 m³
- 10 cm³
Show answer and explanation
Answer: 1000 cm³
A 10 cm cube.
Surface area of a cube with side 3 cm?
- 54 cm²
- 27 cm²
- 9 cm²
- 36 cm²
Show answer and explanation
Answer: 54 cm²
$6 \times 9$ .
Volume of a cylinder, radius 2 cm, height 5 cm, in terms of $\pi$ ?
- $20\pi$ cm³
- $10\pi$ cm³
- $100\pi$ cm³
- $40\pi$ cm³
Show answer and explanation
Answer: $20\pi$ cm³
$\pi \times 4 \times 5$ .
Curved surface area of a cylinder, radius 5 cm, height 4 cm?
- $40\pi$ cm²
- $100\pi$ cm²
- $20\pi$ cm²
- $90\pi$ cm²
Show answer and explanation
Answer: $40\pi$ cm²
$2\pi \times 5 \times 4$ .
Volume of a cone, radius 3 cm, height 7 cm, in terms of $\pi$ ?
- $21\pi$ cm³
- $63\pi$ cm³
- $7\pi$ cm³
- $42\pi$ cm³
Show answer and explanation
Answer: $21\pi$ cm³
$\frac{1}{3} \times 9 \times 7 = 21$ .
Surface area of a sphere with radius 5 cm?
- $100\pi$ cm²
- $25\pi$ cm²
- $\frac{500}{3}\pi$ cm²
- $50\pi$ cm²
Show answer and explanation
Answer: $100\pi$ cm²
$4\pi \times 25$ .

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