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 eight are not formula errors at all. They are decisions — which dimension, which height, which faces, which radius — 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.

• 5.1 — 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.
• 5.1 — 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.
• 5.2 — find the perimeter and area of a rectangle, triangle, parallelogram and trapezium, identifying the perpendicular height rather than a sloping side.
• 5.2 — 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.
• 5.3 — calculate circumference and area of a circle from either the radius or the diameter.
• 5.3 — calculate the length of an arc and the area of a sector for a given central angle, for both the minor and the major region, and find a sector perimeter as an arc plus two radii.
• 5.3 — find the area of a segment as a sector minus a triangle, and give an answer either exactly in terms of \(\pi\) or as a decimal to 3 significant figures, as instructed.
• 5.4 — 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.
• 5.4 — 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.
• 5.5 — 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.
• 5.5 — find the volume and surface area of a frustum by treating the removed cone as similar to the complete cone and subtracting.
• Across the topic — choose the correct dimension, keep intermediate values unrounded, and finish with the correct linear, squared or cubed unit.

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.

Surface Area and Volume

Volume is the amount of space a solid occupies, measured in cubed units, and surface area is the total area of the faces enclosing it, measured in squared units. A cuboid has volume lwh and surface area 2(lw + lh + wh). A prism is any solid with a uniform cross-section, so its volume is the cross-sectional area multiplied by its length and its surface area is two cross-sectional ends plus the lateral faces formed along the cross-section perimeter. A cylinder has volume pi r squared h and curved surface area 2 pi r h, with 2 pi r squared more for the two ends. A pyramid has volume one third of the base area times the perpendicular height, and its surface area is the base plus its triangular faces. A cone has volume one third pi r squared h and curved surface area pi r l, where l is the slant height and, for a right cone, l squared equals r squared plus h squared. A sphere has volume four thirds pi r cubed and surface area 4 pi r squared, and a hemisphere has half that volume, a curved surface of 2 pi r squared and a total surface of 3 pi r squared when its flat circular face is exposed.

Units of Measure

Metric units express how much of a quantity has been measured, and each quantity has its own dimension. A length uses a single conversion factor, an area uses that factor squared and a volume uses it cubed, because an area is a product of two lengths and a volume a product of three. In the metric system 1 m = 100 cm = 1000 mm and 1 km = 1000 m, so 1 square metre is 10 000 square centimetres and 1 cubic metre is 1 000 000 cubic centimetres. Capacity connects to volume through 1 cubic centimetre being 1 millilitre, 1000 cubic centimetres being 1 litre and 1 cubic metre being 1000 litres, and mass through 1000 grams being 1 kilogram.

Circles, Arcs and Sectors

A circle of radius r has circumference 2 pi r, equal to pi d for diameter d, and area pi r squared. An arc is a part of the circumference and a sector is the region enclosed by an arc and the two radii at its ends. For a central angle theta measured in degrees, both are the same fraction theta over 360 of the whole circle, so the arc length is theta over 360 times 2 pi r and the sector area is theta over 360 times pi r squared. The smaller region is the minor sector and the larger is the major sector, whose angle is 360 minus theta. The perimeter of a sector is its arc length plus two radii. A segment is the region between a chord and its arc, and its area is found as the area of the sector minus the area of the triangle formed by the two radii and the chord.

Area and Perimeter

Perimeter is the total length of the boundary enclosing a two-dimensional shape and is measured in linear units; area is the amount of surface the shape encloses and is measured in squared units. A rectangle of length l and width w has perimeter 2(l + w) and area lw, a triangle has area half the base times the perpendicular height, a parallelogram has area base times perpendicular height, and a trapezium with parallel sides a and b separated by perpendicular distance h has area half of a plus b, times h. Each area formula requires a perpendicular height rather than a sloping side. The area of a compound shape is found by splitting it into non-overlapping known regions and adding, or by enclosing it in a complete shape and subtracting the missing part, while its perimeter is traced around the outside boundary only.

Compound Shapes and Parts of Shapes

A compound shape or solid is one built from, or left over from, shapes whose measurements are already known. Its area or volume is found by splitting it into non-overlapping known parts and adding, or by enclosing it in a complete shape and subtracting the part that is missing. Its perimeter is traced around the outside boundary only, and internal cutting lines are excluded. For a compound solid the volumes of the parts simply add, but the surface areas do not: a face where two parts are joined lies inside the solid and is excluded from the external surface area, while a face newly exposed by a cut must be included. A frustum is the solid left when a cone is cut parallel to its base and the top removed; the removed cone is similar to the complete cone, so the height scale factor gives every missing radius and slant length, and the frustum is found by subtracting the small cone from the complete one.

• “\(1\,\mathrm{m^2}=100\,\mathrm{cm^2}\), 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, 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.

• Both papers can assess every part of Topic 5. What changes between them is not the mathematics but the form the answer must take, and marks are lost far more often on the form than 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. If neither is stated on Paper 2, a 3-significant-figure decimal is the normal expectation.
• 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.
• Do not measure the diagram. Diagrams in this topic are frequently labelled “not drawn accurately”. Use the given lengths, never a ruler.
• Answer the question that was asked. Curved surface area, total surface area and volume are three different questions about the same cylinder.