Surface area and volume | OCR GCSE Maths Notes

Visual model

Volume fills a solid, surface area covers its outside

insidesurface areaadd outside facesvolumespace inside

Identify the 3D shape first.

Volume fills the shape.

Surface area covers the outside faces.

Gold-standard guide

20 mins

What you will learn

Calculate measurements for prisms and curved solids.

Use a clear step-by-step method for surface area and volume.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Volume of cylinder = $\pi$ *r²*h

Step 1

Calculate the volume

V = $\pi$ × 4² × 9 = $\pi$ × 16 × 9 = $144pi$ ≈ 452 cm³ (3 s.f.)

Step 2

Calculate the curved surface area

CSA = 2 × $\pi$ × 4 × 9 = $72pi$

Step 3

Add the two circular ends for total surface area

2 × $\pi$ × 4² = $32pi$

Watch out

Students forget to add the circular ends for total surface area — the curved surface area alone is not the total

Cylinder volume

$V = \pi r^{2} h.$

Curved surface

$curved area = 2pi r h.$

Worked example

A cylinder has radius 4 cm and height 9 cm. Find its volume and total surface area. Give answers to 3 significant figures.

Calculate the volume: V = $\pi$ × 4² × 9 = $\pi$ × 16 × 9 = $144pi$ ≈ 452 cm³ (3 s.f.).

Calculate the curved surface area: CSA = 2 × $\pi$ × 4 × 9 = $72pi$ .

Add the two circular ends for total surface area: 2 × $\pi$ × 4² = $32pi$ . TSA = $72pi$ + $32pi$ = $104pi$ ≈ 327 cm² (3 s.f.).

Final answer

V ≈ 452 cm³; TSA ≈ 327 cm²

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

A cylinder has radius 4 cm and height 9 cm. Find its volume and total surface area. Give answers to 3 significant figures.

3 marks4 minssurface-area-and-volume-worked

Show solution

Worked solution

1.Calculate the volume: V = $\pi$ × 4² × 9 = $\pi$ × 16 × 9 = $144pi$ ≈ 452 cm³ (3 s.f.).
2.Calculate the curved surface area: CSA = 2 × $\pi$ × 4 × 9 = $72pi$ .
3.Add the two circular ends for total surface area: 2 × $\pi$ × 4² = $32pi$ . TSA = $72pi$ + $32pi$ = $104pi$ ≈ 327 cm² (3 s.f.).

Final answer

V ≈ 452 cm³; TSA ≈ 327 cm²

Mark points

M1: calculate the volume
M1: calculate the curved surface area
M1: add the two circular ends for total surface area
A1: V ≈ 452 cm³; TSA ≈ 327 cm²

Watch out

Students forget to add the circular ends for total surface area — the curved surface area alone is not the total.Check whether the question says 'curved surface area' or 'total surface area' and include the relevant faces.

Diagnosticrecall

Find the volume of a cone with radius 6 cm and height 8 cm. Give in terms of $\pi$ .

1 mark2 minssurface-area-and-volume-q1

Show solution

Worked solution

1.Spot the skill: Volume of cylinder = $\pi$ *r²*h.
2.Use the calculate the volume stage first, then calculate the curved surface area.
3.Keep the final answer visible: $96pi c$ m³.

Final answer

$96pi c$ m³

Mark points

M1: use the correct volume of cylinder = $\pi$ *r²*h. curved surface area = 2* $\pi$ *r*h.total surface area = 2* $\pi$ *r*h + 2* $\pi$ *r² (add the two circular ends). volume of cone = ( $\frac{1}{3}$ )* $\pi$ *r²*h.surface area of sphere = 4* $\pi$ *r². volume of sphere = ( $\frac{4}{3}$ )* $\pi$ *r³.
A1: $96pi c$ m³

Watch out

Easyprocedure

Find the surface area of a sphere with radius 5 cm, to 1 d.p.

2 marks3 minssurface-area-and-volume-q2

Show solution

Worked solution

1.Spot the skill: Volume of cylinder = $\pi$ *r²*h.
2.Use the calculate the curved surface area stage first, then add the two circular ends for total surface area.
3.Keep the final answer visible: 314.2 cm².

Final answer

314.2 cm²

Mark points

M1: use the correct volume of cylinder = $\pi$ *r²*h. curved surface area = 2* $\pi$ *r*h.total surface area = 2* $\pi$ *r*h + 2* $\pi$ *r² (add the two circular ends). volume of cone = ( $\frac{1}{3}$ )* $\pi$ *r²*h.surface area of sphere = 4* $\pi$ *r². volume of sphere = ( $\frac{4}{3}$ )* $\pi$ *r³.
A1: 314.2 cm²

Watch out

Mediumreasoning

A cylinder has volume $200pi c$ m³ and radius 5 cm. Find its height.

3 marks4 minssurface-area-and-volume-q3

Show solution

Worked solution

1.Spot the skill: Volume of cylinder = $\pi$ *r²*h.
2.Use the add the two circular ends for total surface area stage first, then calculate the volume.
3.Keep the final answer visible: 8 cm.

Final answer

8 cm

Mark points

M1: use the correct volume of cylinder = $\pi$ *r²*h. curved surface area = 2* $\pi$ *r*h.total surface area = 2* $\pi$ *r*h + 2* $\pi$ *r² (add the two circular ends). volume of cone = ( $\frac{1}{3}$ )* $\pi$ *r²*h.surface area of sphere = 4* $\pi$ *r². volume of sphere = ( $\frac{4}{3}$ )* $\pi$ *r³.
A1: 8 cm

Watch out

Hardproblem solving

Find the volume of a hemisphere with diameter 12 cm. Give in terms of $\pi$ .

3 marks5 minssurface-area-and-volume-q4

Show solution

Worked solution

1.Spot the skill: Volume of cylinder = $\pi$ *r²*h.
2.Use the calculate the volume stage first, then calculate the curved surface area.
3.Keep the final answer visible: $144pi c$ m³.

Final answer

$144pi c$ m³

Mark points

M1: use the correct volume of cylinder = $\pi$ *r²*h. curved surface area = 2* $\pi$ *r*h.total surface area = 2* $\pi$ *r*h + 2* $\pi$ *r² (add the two circular ends). volume of cone = ( $\frac{1}{3}$ )* $\pi$ *r²*h.surface area of sphere = 4* $\pi$ *r². volume of sphere = ( $\frac{4}{3}$ )* $\pi$ *r³.
A1: $144pi c$ m³

Watch out

Exam-stylemulti-step

A cone has slant height 13 cm and base radius 5 cm. Find its total surface area.

4 marks6 minssurface-area-and-volume-q5

Show solution

Worked solution

1.Spot the skill: Volume of cylinder = $\pi$ *r²*h.
2.Use the calculate the curved surface area stage first, then add the two circular ends for total surface area.
3.Keep the final answer visible: $90pi c$ m².

Final answer

$90pi c$ m²

Mark points

M1: use the correct volume of cylinder = $\pi$ *r²*h. curved surface area = 2* $\pi$ *r*h.total surface area = 2* $\pi$ *r*h + 2* $\pi$ *r² (add the two circular ends). volume of cone = ( $\frac{1}{3}$ )* $\pi$ *r²*h.surface area of sphere = 4* $\pi$ *r². volume of sphere = ( $\frac{4}{3}$ )* $\pi$ *r³.
A1: $90pi c$ m²

Watch out

Grade 9 stretchproblem solving

Find the total surface area of a closed cylinder with radius 3 cm and height 10 cm. Give the answer in terms of $\pi$ .

r=3

h=10

2\pi rh+2\pi r^{2}

4 marks7 minssurface-g9

Show solution

Worked solution

1.Find the area of both circular ends.
2.Find the curved surface area.
3.Add all surfaces.

Final answer

$78pi c$ m²

Mark points

M1: use $2pi$ (3²)
M1: use $2pi$ (3)(10)
A1: $78pi c$ m²

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Surface area and volume - 2 marksFind the volume of a cone with radius 6 cm and height 8 cm. Give in terms of

\pi

.Mark answer

Answer

$96pi c$ m³

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer

Answer

Rectangle, square

4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer

Answer

42 cm

Mastery check

I can explain the method for surface area and volume.
I can show clear working without skipping key steps.
I can avoid this mistake: Students forget to add the circular ends for total surface area — the curved surface area alone is not the total.Check whether the question says 'curved surface area' or 'total surface area' and include the relevant faces.

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Explain surface area and volume from the beginning.Give me a hint for a surface area and volume question.Quiz me on surface area and volume.

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