Good Learning Co.
SavannahYear 8
← Maths
MathsGCSE maths: geometry, Pythagoras and trigonometry30 min★★★ difficulty

Area, surface area and volume

Mensuration questions are rarely about one formula. They usually combine shapes, so the skill is breaking a solid into parts you already know.

Lesson overview

What you'll learn in this lesson

Reason geometrically and calculate lengths, angles, areas and volumes.

Key learning points

  • • Circles
  • • Prisms and cylinders
  • • Compound solids and units

This lesson at a glance

  • 30 minutes
  • 16 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

reasongeometricallycalculatelengthsangles

Scroll down — the lesson carries on below

Part 1 of 16 · Discover6%
1

Watch & discover

Part 1 of 16

Watch first

Area, surface area and volume

An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.

In a nutshell

Reason geometrically and calculate lengths, angles, areas and volumes.

2

Learning cycle

Part 2 of 16

Learning cycle 1 of 2

Part 1 · Circles

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 16

Learn

Circles

Circumference = πd and area = πr². For a sector, take the fraction θ ÷ 360 of the whole circle. Arc length uses the circumference and sector area uses the area — mixing them is the most common error, so write the fraction first and the full-circle formula second.

4

Explore the idea

Part 4 of 16

Learn

Prisms and cylinders

The volume of any prism is the cross-sectional area multiplied by the length. A cylinder is a prism with a circular cross-section, so its volume is πr²h. Its surface area is two circles plus a rectangle whose width is the circumference: 2πr² + 2πrh.

5

Quick check

Part 5 of 16

Quick check

The area of a circle is

6

Quick check

Part 6 of 16

Quick check

The volume of a prism is

7

Reset break

Part 7 of 16

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
8

Learning cycle

Part 8 of 16

Learning cycle 2 of 2

Part 2 · Compound solids and units

A short piece of teaching, then a check to make sure it has landed.

9

Explore the idea

Part 9 of 16

Learn

Compound solids and units

Break a compound solid into standard parts, calculate each and add or subtract. Keep answers in terms of π until the final step to avoid rounding error. Check units: area in cm², volume in cm³, and remember 1 litre = 1,000 cm³ when a question switches to capacity.

10

Quick check

Part 10 of 16

Quick check

A 90° sector is what fraction of the circle?

11

Quick check

Part 11 of 16

Quick check

1 litre equals

12

Explore the idea

Part 12 of 16

Worked example

Worked answer: find the volume of a cylinder of radius 4 cm and height 10 cm (3 marks)

Volume of a cylinder = πr²h (1). Substituting: π × 4² × 10 = 160π (1). Evaluating gives 502.65 cm³, which is 503 cm³ to 3 significant figures (1). Using the diameter instead of the radius quadruples the answer, so check which the question gives before substituting.

13

Challenge round

Part 13 of 16

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

14

Challenge round

Part 14 of 16

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

Break a compound solid into standard parts, ____ each and add or subtract.

15

Mastery quiz

Part 15 of 16

Marked quiz

End of lesson quiz: Area, surface area and volume

4 questions, marked with the reasoning shown. No timer.

  1. 1. The area of a circle is

  2. 2. The volume of a prism is

  3. 3. A 90° sector is what fraction of the circle?

  4. 4. 1 litre equals

16

Lesson round-up

Part 16 of 16

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 16 · Watch & discover

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