On the Foundations Check, area, perimeter, volume and units came back secure to step 6 and broke at step 7. Step 7 is the hardest Extended rung, so this is one of your stronger strands — the foundations are in place up to Extended level, and only the top rung gave way.
That is genuinely good news, and it changes what this guide should do. It is not a rebuild from primary level. It is a tightening of three specific things that break at the top rung:
Your check also found sign errors recurring across topics. In mensuration they appear when a piece is removed from a shape: the subtraction at the end goes the wrong way, or the hole gets added instead of taken off. Section 5 deals with it directly.
| Code | Sub-topic | Expected difficulty for you |
|---|---|---|
| E5.1 | Units of measure | Medium — the area and volume trap |
| E5.2 | Area and perimeter | High — rebuilt carefully here |
| E5.3 | Circles, arcs and sectors | Medium — answers left in terms of π |
| E5.4 | Surface area and volume | Medium — a formula list to own |
| E5.5 | Compound shapes and parts of shapes | Medium — more steps, not harder ideas |
Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. So every answer in this guide is left exact — in terms of π, or as a surd — rather than turned into a decimal.
Number sense and the four operations: solid on all seven rungs. No gap anywhere, right up to Extended. Mensuration is mostly multiplication, halving, and one subtraction at the end — which is exactly the machinery the check says you already have.
So the marks here are not lost to arithmetic. They are lost to three things: converting an area as if it were a length, using a slanted side as a perpendicular height, and forgetting which formulae are printed on the paper. All three are fixable in an evening, and none of them is about being good at maths.
Every idea appears four times, with less help each time.
Where an answer contains π, type it as pi — so 12pi means 12π, and
8pi+16 means 8π + 16. Section 6 is a mixed, unlabelled set.
Length conversions almost never go wrong. Area and volume conversions go wrong constantly, and for one reason: people convert the number instead of converting the square.
| Quantity | Conversions to know |
|---|---|
| Length | 10 mm = 1 cm · 100 cm = 1 m · 1000 m = 1 km |
| Mass | 1000 mg = 1 g · 1000 g = 1 kg · 1000 kg = 1 tonne |
| Capacity | 1000 ml = 1 litre · 100 cl = 1 litre |
| Volume and capacity together | 1 ml = 1 cm³ · 1 litre = 1000 cm³ · 1 m³ = 1000 litres |
This is the highest-risk sub-topic in the whole of topic 5, and not because it is hard. It is high risk because only one of these formulae is printed on the paper, so three of the four have to come out of your head under exam pressure.
| Shape | Area formula | On the paper? |
|---|---|---|
| Rectangle | length × width | No — you supply it |
| Triangle | ½ × base × perpendicular height | Yes — given |
| Parallelogram | base × perpendicular height | No — you supply it |
| Trapezium | ½(a + b) × h | No — you supply it |
It is worth noticing what that list implies: the trapezium, the one most people forget, is not handed to you, so learn ½(a + b)h by heart. The triangle is on the formula list, but you use it so often that you should know it anyway. After that, the risk is not memory, it is reading the diagram.
Perimeter is the distance all the way round: add the sides, answer in cm. Area is the space inside: answer in cm². They are different questions and they carry different units.
Two formulae for the whole circle, then everything else is a fraction of them. On Paper 2 the answers stay in terms of π, which is not a cop-out — it is the exact answer, and it is what the mark scheme wants.
| What | Formula | Notice |
|---|---|---|
| Circumference | C = 2πr = πd | Uses the radius once — a length, so the answer is in cm |
| Area | A = πr² | Uses the radius twice — an area, so cm² |
| Arc length | (θ ÷ 360) × 2πr | The fraction of the way round |
| Sector area | (θ ÷ 360) × πr² | The same fraction of the whole area |
An arc and a sector are the same fraction of the circle, so the fraction is worked out once and used twice. Simplify it before multiplying: 60/360 is 1/6, and a sixth of something is far easier than 60 lots divided by 360.
r = 6 on its own line so the halving is
visible and cannot be skipped.8π + 16. Leave it in that form. Do not turn it into a decimal, and do not try
to add the 8π to the 16 — they are different kinds of term, exactly like 8x + 16.A formula list, and one idea that ties most of it together: a prism has the same cross-section all the way along, so its volume is that cross-section multiplied by its length. A cuboid is a prism. A cylinder is a prism with a circular cross-section. Learning it that way turns three formulae into one.
| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | l × w × h | 2(lw + lh + wh) |
| Any prism | cross-section area × length | 2 × cross-section + perimeter × length |
| Cylinder | πr²h | curved 2πrh, total 2πrh + 2πr² |
| Sphere | 4⁄3πr³ | 4πr² |
| Cone | ⅓πr²h | curved πrl, total πrl + πr² |
| Pyramid | ⅓ × base area × height | base + the triangular faces |
h is the vertical height, straight up the middle,
and it is the one in the volume. l is the slant height, down the sloping side, and it is the one in
the curved surface area. They are linked by Pythagoras: r² + h² = l². Questions routinely give you
one and want the other, and the triangle is nearly always 3-4-5 or 6-8-10.Volume = ⅓ × base area × perpendicular height. The surface area is the base plus every triangular face, and each triangle needs its slant height: the height up the middle of that face. Find it with Pythagoras from the perpendicular height and half the base edge.
Nothing new is being taught here. A compound shape is two or three shapes you already know, joined together or with a piece taken out. What makes it worth its own section is that every extra step is another place to slip, and your check flagged sign errors in exactly this kind of multi-step work.
whole − hole, and the answer must be smaller than the whole. If your answer for a
shape with a hole in it is bigger than the shape without the hole, the subtraction went the wrong way. That
one check catches the sign error every time.Perimeter is the walk round the outside. If two pieces are joined, the join is inside the shape, so it is not part of the walk. This is the most common perimeter error: including the join.
The 6 cm end where the semicircle sits does not appear, because the curve replaced it. The 6 cm at the other end does.
A minor segment is the piece cut off by a chord. It is the sector with the triangle taken away: segment = sector − triangle, where the triangle is made by the two radii and the chord. Its area is ½r² sin θ, which is just ½r² when θ = 90°. The major segment is the whole circle minus the minor one.
Volume: add the pieces, or subtract a hole. Surface area: only the faces on the outside count. Where two pieces are joined, the circle (or face) between them is hidden inside and is left out.
A frustum is a cone (or pyramid) with its top cut off parallel to the base. Work it as big cone − small cone. The small cone is similar to the big one, so its radius and height come from the scale factor. The curved surface is πRL − πrl; the total surface area adds the top circle πr² and the base πR².
Fifteen questions from everything above, shuffled and unlabelled. The skill being tested is choosing the
method, which is the part ordinary revision quietly does for you. Before each one, name the shape and say which
formula it needs. Type π as pi.
Six things, and between them they cover most of what an examiner can ask in topic 5:
r = ... on its own line so the step cannot be skipped.Come back to section 6 in a week without reading anything above it. A high score on a cold run means this topic is set.