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
- 20 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 20
Visual introductionPicture this
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.
In a nutshell
Reason geometrically and calculate lengths, angles, areas and volumes.
Learning cycle
Part 2 of 20
Learning cycle 1 of 2
Part 1 · Circles
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 20
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.
Reset break
Part 4 of 20
Pause
That's sitting 1 of 5 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.
Explore the idea
Part 5 of 20
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.
Quick check
Part 6 of 20
Quick check
Part 7 of 20
Reset break
Part 8 of 20
Pause
That's sitting 2 of 5 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.
Learning cycle
Part 9 of 20
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.
Explore the idea
Part 10 of 20
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.
Quick check
Part 11 of 20
Reset break
Part 12 of 20
Pause
That's sitting 3 of 5 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.
Quick check
Part 13 of 20
Explore the idea
Part 14 of 20
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.
Challenge round
Part 15 of 20
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
Reset break
Part 16 of 20
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 20
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.
Challenge round
Part 18 of 20
Game · Recall cards
The area of a circle is
Card 1 of 4
Mastery quiz
Part 19 of 20
Marked quiz
End of lesson quiz: Area, surface area and volume
4 questions, marked with the reasoning shown. No timer.
1. The area of a circle is
2. The volume of a prism is
3. A 90° sector is what fraction of the circle?
4. 1 litre equals
Lesson round-up
Part 20 of 20
Lesson round-up
Ready when you are
Quiz score
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Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaPart 1 of 20 · Watch & discover