GCSE Maths Mensuration revision

Metric conversions are powers of ten: 1 m = 100 cm, 1 km = 1000 m. Areas square the length factor — 1 m² = 100² = 10 000 cm² — and volumes cube it: 1 m³ = 100³ = 1 000 000 cm³. Capacity joins through 1 litre = 1000 cm³, so a 0.75-litre bottle holds 750 cm³. Mixing these up is the costliest mensuration slip: an answer “in cm²” that was computed with metres is wrong by a factor of 10 000, not 100.

Perimeter is the total distance around a shape — add every edge, including the ones you must deduce. L-shapes keep a secret: an L cut from a rectangle has the same perimeter as the full rectangle (the notch pushes one horizontal and one vertical edge inward without changing total lengths), while its area drops by the missing rectangle. A 8 cm × 5 cm rectangle with a 3 cm × 2 cm corner removed has perimeter 2(8 + 5) = 26 cm and area 40 − 6 = 34 cm².

GCSE papers do not print a formula list, so three area formulas live in memory. Triangle: ½ × base × height — the height meets the base at right angles and may fall outside the shape for obtuse triangles. Parallelogram: base × height (it is a sheared rectangle). Trapezium: ½(a + b)h, the two parallel sides averaged, times the perpendicular height between them: parallel sides 7 and 11 with height 6 give ½ × 18 × 6 = 54 cm².

Composite shapes split into rectangles, triangles and trapezia — or, for a frame or ring, subtract the hole from the outer shape. Always mark the dimensions you derive (the far side of an L, the height outside a triangle) on the diagram before calculating; two marks routinely sit on those deductions.

Every circle fact hangs off the radius r: circumference C = 2πr = πd and area A = πr². Keep them distinct by their units — circumference behaves like a length, area like paint. A radius of 6 cm gives C = 12π ≈ 37.7 cm and A = 36π ≈ 113.1 cm². Use the calculator’s π button throughout; early rounding to 3.14 drifts answers out of tolerance.

A sector is a pizza slice of angle θ°, and θ⁄360 is the fraction of the whole circle it represents — of the circumference (the arc) and of the area alike. A 120° sector of a radius-9 circle takes a third: arc = ⅓ × 18π = 6π ≈ 18.8 cm, area = ⅓ × 81π = 27π ≈ 84.8 cm². Reverse questions — find the angle from the arc length — divide by the whole instead of multiplying.

A prism is a 3D shape whose slice stays the same along its length: volume = area of cross-section × length. A cuboid is the simple case (3 × 4 × 10 = 120); a trapezium-shaped trough with cross-section 90 cm² and length 25 cm holds 90 × 25 = 2250 cm³. A cylinder is a circular prism: V = πr²h — a radius-5, height-12 cylinder holds 300π ≈ 942.5 cm³. Surface areas are assembled from flat faces and curves: a cylinder’s total is 2πr² (the two caps) plus 2πrh (the curved wall); a closed 4×3×5 cuboid needs 2(12 + 20 + 15) = 94 cm².

Higher-tier solids come with formulas printed in the question — you apply, not recall. Sphere of radius 3: V = (4⁄3)π(27) = 36π ≈ 113.1 cm³ and S = 4π(9) = 36π ≈ 113.1 cm² (equal here — a curiosity of r = 3). Cone of radius 4, height 3: the slant height l comes from Pythagoras (√(3² + 4²) = 5), then V = (1⁄3)π(16)(3) = 16π ≈ 50.3 cm³ and the curved surface is πrl = 20π ≈ 62.8 cm². Composite solids split into parts — a cylinder with a hemispherical lid needs half the sphere’s volume added, and (for surface area) half the sphere’s curved skin plus the exposed circle, not the hidden one.