1MA1 · G16–G17Geometry & MeasuresFoundation & HigherGrades 4–8

Volume & Surface Area

Volume measures the space inside a 3D shape; surface area measures the total area of its faces. For prisms and cylinders one idea covers everything: volume is the area of the cross-section times the length. Pyramids, cones and spheres need their own formulas; the cone and sphere ones are given in the question when needed, but you still need to know how to use them.

55 min Mensuration

The big picture

A prism has the same cross-section all the way through, so its volume stacks that area along the length. A cylinder is just a prism with a circular cross-section. Pyramids and cones take up exactly one third of the prism that encloses them, which is where the $\tfrac13$ in their formulas comes from.

Higher-tier questions often melt one solid and recast it as another — the volume stays the same, so you set the two volume expressions equal and solve. Surface area questions reward sketching a net so no face is missed.

What you'll learn

Calculate the volume and surface area of cuboids and prisms
Calculate the volume and surface area of cylinders
Use formulas for the volume and surface area of pyramids, cones and spheres (Higher)
Solve problems involving composite solids and equal volumes

Prisms and cylinders

Volume of a prism $=$ area of cross-section × length. For a cuboid, that is length × width × height.

A cylinder of radius $r$ and height $h$ has volume $\pi r^{2}h$ .

Volume units are cubed: cm³, m³. $1$ m³ $= 1\,000\,000$ cm³, and $1$ litre $= 1000$ cm³.

V = Al

Prism: cross-section area × length.

V = \pi r^{2}h

Cylinder.

Example 1A cylinder has radius 4 cm and height 10 cm. Find its volume and total surface area in terms of

\pi

1Volume

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

cm³.

2Two circular ends:

2 \times 16\pi = 32\pi

. Curved surface:

2\pi \times 4 \times 10 = 80\pi

AnswerVolume

160\pi

cm³; surface area

112\pi

cm²

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

Find the volume of a cylinder with radius 3 cm and height 8 cm, in terms of $\pi$ .

Your answer

Type π as pi, e.g. 40pi.

Show worked solution

Solution

1 $V = \pi \times 3^2 \times 8 = 72\pi$ cm³.

Answer: $72\pi$ cm³

Why this works

The curved surface of a cylinder unrolls into a rectangle: one side is the height, the other is the circumference $2\pi r$ . That is why its area is $2\pi rh$ .

Surface area

Surface area is the total area of all faces. Sketch the net, list every face, and add.

A cuboid has three pairs of identical rectangular faces: $2(lw + lh + wh)$ .

For a triangular prism, find the two triangular ends and the three rectangles.

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

Find the surface area of a cuboid measuring 5 cm by 4 cm by 2 cm, in cm².

Your answer

Show worked solution

Solution

1Face pairs: $20$ , $10$ and $8$ cm².
2 $2(20 + 10 + 8) = 76$ cm².

Answer: $76$ cm²

Tip — Surface area is measured in square units (cm²), volume in cubic units (cm³). Mixing them up is an easy mark to lose.

Pyramids, cones and spheres

Pyramid: $V = \tfrac13 \times$ base area × perpendicular height.

Cone: $V = \tfrac13\pi r^{2}h$ ; curved surface area $= \pi rl$ , where $l$ is the slant height. Pythagoras links them: $l^{2} = r^{2} + h^{2}$ .

Sphere: $V = \tfrac43\pi r^{3}$ ; surface area $= 4\pi r^{2}$ . A hemisphere is half a sphere — remember the flat circular face when finding its total surface area.

V = \tfrac13\pi r^{2}h,\ \ S = \pi rl

Cone (curved surface).

V = \tfrac43\pi r^{3},\ \ S = 4\pi r^{2}

Sphere.

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

\pi

1Volume

= \tfrac13 \times \pi \times 36 \times 8 = 96\pi

cm³.

2Slant height

l = \sqrt{36 + 64} = 10

cm.

3Total surface area

= \pi \times 6 \times 10 + \pi \times 36 = 60\pi + 36\pi

AnswerVolume

96\pi

cm³; surface area

96\pi

cm²

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

Find the volume of a sphere with radius 3 cm, in terms of $\pi$ .

Your answer

Type π as pi, e.g. 40pi.

Show worked solution

Solution

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

Answer: $36\pi$ cm³

Composite solids and equal volumes

For a composite solid, find each part’s volume and add (or subtract for a hollow).

When a solid is melted and recast, the volume is unchanged. Set the volumes equal and solve for the unknown dimension.

Example 1A solid metal sphere of radius 6 cm is melted and recast into a cylinder of radius 4 cm. Find the height of the cylinder.

1Sphere volume

= \tfrac43\pi \times 216 = 288\pi

2Cylinder:

\pi \times 16 \times h = 288\pi

16h = 288

Answer

h = 18

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

A cube of side 6 cm is melted and recast as a cuboid with base 9 cm by 4 cm. Find the height of the cuboid, in cm.

Your answer

Show worked solution

Solution

1Volume $= 216$ cm³.
2Height $= 216 \div 36 = 6$ cm.

Answer: $6$ cm

Tip — In equal-volume problems, cancel $\pi$ from both sides straight away. The remaining arithmetic is usually neat.

Think like an examiner

Write the formula and substitution before calculating.
List faces for surface area, including hidden ones.
Give correct units: cm³ for volume, cm² for area.

Volume and surface area

V = \pi r^{2}h

Cylinder volume.

V = \tfrac13 Ah

Pyramid or cone.

V = \tfrac43\pi r^{3}

Sphere volume.

S = 4\pi r^{2}

Sphere surface area.

Watch out for these

Using the slant height in the cone volume formula.

Volume uses the perpendicular height.

Forgetting the flat face of a hemisphere.

Total surface area includes the circular base

\pi r^{2}

Converting 1 m³ to 100 cm³.

1

m³

= 1\,000\,000

cm³.

Forgetting the

\tfrac13

for pyramids and cones.

They are one third of the enclosing prism.

Stretch yourself

A cylindrical glass of internal radius 3 cm is filled with water to a depth of 8 cm. A solid metal sphere of radius 1.5 cm is dropped in and sinks. By how much does the water level rise?

Hint — The rise in water volume equals the volume of the sphere. The extra water forms a cylinder of radius 3 cm.

Reveal solution

SolutionSphere volume

= \tfrac43\pi \times 1.5^{3} = \tfrac43\pi \times 3.375 = 4.5\pi

cm³. The water rises by

h

in a cylinder of radius 3:

9\pi h = 4.5\pi

, so

h = 0.5

cm. The sphere is fully covered, since the new depth of 8.5 cm exceeds its diameter of 3 cm.

\blacksquare

Questions students ask

Are the cone and sphere formulas given?

Edexcel includes the cone and sphere formulas in the question where they are needed. Learn the prism, cylinder and pyramid formulas yourself.

What is the difference between slant height and height?

Height is perpendicular from base to apex; slant height is measured along the sloping surface.

How many cm³ are in a litre?

1000 cm³ = 1 litre.

Key takeaways

Prism volume $=$ cross-section area × length; cylinder $\pi r^{2}h$ .
Surface area: add all faces; cylinder curved surface $2\pi rh$ .
Pyramids and cones use $\tfrac13$ ; cone curved area $\pi rl$ .
Sphere $\tfrac43\pi r^{3}$ and $4\pi r^{2}$ .
Melted and recast solids keep the same volume.

Exam-style questions

Practise it the way it is examined

3 original questions · 6 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 41 mark

A prism has a cross-section of area 15 cm² and length 12 cm. Work out its volume.[1]
Show mark scheme
- B1 $180$ cm³
Answer: $180$ cm³

Question 2Grade 62 marks

A cone has radius 5 cm and perpendicular height 12 cm. Work out its curved surface area in terms of $\pi$ . [Curved surface area $= \pi r l$ ][2]
Show mark scheme
- M1Slant height $\sqrt{25 + 144} = 13$
- A1 $65\pi$ cm²
Answer: $65\pi$ cm²

Question 3Grade 83 marks

A solid is a hemisphere of radius 3 cm on top of a cylinder of radius 3 cm and height 10 cm. Work out its volume in terms of $\pi$ .[3]
Show mark scheme
- M1Cylinder $90\pi$
- M1Hemisphere $\frac{1}{2} \times \frac{4}{3}\pi \times 27 = 18\pi$
- A1 $108\pi$ cm³
Answer: $108\pi$ cm³

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.

Next step

Keep practising

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