Mensuration

• Length, area, volume: one factor, squared factor, cubed factor. Decide the dimension before you reach for a formula, and the units look after themselves.
• Six of these nine are not formula errors at all. They are decisions — which dimension, which height, which faces, which radius, which region — taken too quickly. Slow the first thirty seconds of every mensuration question and most of this list disappears.
• Decide the dimension. Convert once, at the start. Split the shape. Keep \(\pi\) exact. Count only the faces you could paint. Attach the unit.
• Two additions, one sentence: any angle including the major one, and the frustum by similarity and subtraction.

• C5.1 — convert between millimetres, centimetres, metres and kilometres, between \(\mathrm{mm^2}\), \(\mathrm{cm^2}\), \(\mathrm{m^2}\) and \(\mathrm{km^2}\), and between \(\mathrm{mm^3}\), \(\mathrm{cm^3}\) and \(\mathrm{m^3}\), in both directions, and explain why the factor is squared or cubed.
• C5.1 — move between volume and capacity using \(1\,\mathrm{cm^3}=1\text{ ml}\), \(1000\,\mathrm{cm^3}=1\text{ litre}\) and \(1\,\mathrm{m^3}=1000\text{ litres}\), and between grams and kilograms.
• C5.2 — find the perimeter and area of a rectangle, triangle, parallelogram and trapezium, identifying the perpendicular height rather than a sloping side.
• C5.2 — deduce missing lengths on a compound diagram, then find its area by adding non-overlapping parts or by subtracting a cut-out, and trace its perimeter around the outside boundary only.
• C5.3 — calculate circumference and area of a circle from either the radius or the diameter.
• C5.3 — calculate the length of an arc and the area of a sector as a fraction of the whole circle, for a central angle that is a factor of \(360^\circ\), and find a sector perimeter as an arc plus two radii.
• C5.3, C5.5 — find the area of a segment — a part of a shape — as a sector minus a triangle when the two radii meet at a right angle, and give an answer either exactly in terms of \(\pi\) or as a decimal to 3 significant figures, as instructed.
• C5.4 — calculate the volume and surface area of a cuboid, a prism of any uniform cross-section, a cylinder, a pyramid, a cone, a sphere and a hemisphere.
• C5.4 — build a surface area from a net so that every exposed face is counted exactly once, and tell a perpendicular height apart from a slant height.
• C5.5 — find the perimeter, area, external surface area and volume of a compound shape or compound solid, excluding internal joined faces and including newly exposed cut faces.
• C5.5 — find the volume and the surface area of a part of a solid, such as half a sphere, deciding for yourself which faces are exposed.
• Across the topic — choose the correct dimension, keep intermediate values unrounded, and finish with the correct linear, squared or cubed unit.
• Extended E5.3 — calculate arc length and sector area for any central angle, not only angles that divide \(360^\circ\) exactly.
• Extended E5.3 — work with major sectors and major arcs as well as minor ones, using either \(360^\circ-\theta\) or subtraction from the whole circle.
• Extended E5.5 — find the volume and the surface area of a frustum by treating the removed cone as similar to the complete cone and subtracting.

Why context questions are harder: nothing in them is labelled “radius 30 cm”. You are told the width of a water butt, or that a lawn is a quarter of a circle, and the first job is to turn the description into the dimension the formula wants. Read once for the shape, once for the numbers, and once for the unit the answer must be in.

• “\(1\,\mathrm{m^2}=100\,\mathrm{cm^2}\), because \(1\text{ m}=100\text{ cm}\).” Fix A square metre is a square measuring 100 cm along each edge, so it holds \(100\times100=10\,000\) square centimetres. The length factor is used twice for an area and three times for a volume: \(1\,\mathrm{m^3}=1\,000\,000\,\mathrm{cm^3}\). Test Does your converted number look 10 000 times bigger, not 100 times? If not, you used the linear factor.
• “The 7 cm on the sloping edge is the height of the parallelogram.” Fix Area formulas need the perpendicular height — the shortest distance between the base and the opposite side or vertex. A sloping side is longer than the perpendicular height, so using it always overstates the area. Test Is there a right-angle mark where the height meets the base? If not, that length is not the height.
• “The diameter is 10 cm, so I will use \(r=10\).” Fix Halve the diameter before it goes anywhere near \(\pi r^2\). This single slip multiplies a circle area by four. Test Circle the letter you were given on the diagram — \(r\) or \(d\) — before writing the formula.
• “The perimeter of a sector is its arc length.” Fix A sector is bounded by an arc and two radii. Its perimeter is \(\text{arc}+2r\). The arc alone is only the curved part of the boundary. Test Trace the boundary with a finger. If your finger left the pencil line, you have missed a piece.
• “A cone's curved surface uses its height \(h\).” Fix The curved surface area is \(\pi rl\), where \(l\) is the slant height along the sloping edge. For a right cone, \(l=\sqrt{r^2+h^2}\), so \(l\) is always the larger of the two. Test Volume uses \(h\); curved surface uses \(l\). If a question gives you one and needs the other, Pythagoras is the missing step.
• “The surface area of the whole solid is the sum of the surface areas of its parts.” Fix Where two parts are joined, the shared face is inside the solid and is not a surface at all. Add the faces you could actually paint, and no others. Test For a hemisphere sitting on a cylinder of the same radius, two circles disappear from the total — the top of the cylinder and the flat face of the hemisphere.

• Use the protocol as a checklist under pressure. If a mensuration question stalls, it is almost always because step A has not been done properly. Go back and draw the split.
• Formula-sheet reality: of these four, only the triangle area \(A=\tfrac12bh\) is printed on the paper. The rectangle, parallelogram and trapezium relationships have to be known. The trapezium is the one most often misremembered — it averages the two parallel sides, then multiplies by the gap between them.
• Neither of these two is printed on the paper — on either route's List of formulas — but neither needs to be memorised as a separate fact. If you remember “a fraction of the whole circle” you can rebuild both in the margin from \(C=2\pi r\) and \(A=\pi r^2\), which are printed.
• This is why the net matters. If you can draw the net, you cannot miscount a face — and miscounting faces, rather than misremembering formulas, is what costs marks on surface area questions.
• Every one of these three had the same two-part structure: a formula, then a unit decision. In a real paper the unit decision is worth as much as the formula, and it is the part most often left until it is too late to notice.
• Marking yourself honestly. Award the method marks only if you actually wrote the formula and the substitution before the answer. A correct number with no visible method earns fewer marks in the real thing than it does at your desk.

• All candidates take exactly two components. Which two depends on your route, and the pairing is the thing most often got wrong: each route has one non-calculator paper and one calculator paper, and the paper numbers do not line up the way you might guess.
• Papers 1 and 2 are the non-calculator papers. Not Paper 1 alone. If you are an Extended candidate, your non-calculator paper is Paper 2 and your calculator paper is Paper 4. Both papers on your route can assess every part of Topic 5; a scientific calculator is required for Papers 3 and 4, and algebraic and graphical calculators are not permitted on any paper.
• What changes between a non-calculator paper and a calculator paper is not the mathematics but the form the answer must take, and marks are lost on the form as readily as on the formula.
• Read the instruction, then choose the form. “Give your answer in terms of \(\pi\)” means the symbol \(\pi\) must appear in the answer. “Give your answer correct to 3 significant figures” means it must not. Where a question asks for an exact value, the answer may need to be in terms of \(\pi\) or in surd form. If no accuracy is stated on a calculator paper, a 3-significant-figure decimal is the normal expectation — but an answer that is already exact to four or five significant figures should not be rounded at all.
• Convert before you calculate, not after. Put every measurement in the figure into one unit first; converting a finished area or volume is where squared and cubed factors get forgotten.
• Write the formula down before substituting. On a question worth several marks, a correctly quoted formula with one arithmetic slip still earns method marks.