Mensuration questions are not hard arithmetic - they are formula selection and unit discipline. The examiner gives you a shape, some lengths, and a unit; the marks are lost by using the diameter as the radius, by mixing centimetres into a metre formula, or by choosing area when the question asks for volume. Knowing the formulas cold removes all three risks.
These same formulas reappear in physics, in everyday shopping, and in any later question about capacity or material.
The model omits composite shapes, which are handled by splitting them into the standard pieces and adding or subtracting. It also assumes regular figures; real irregular shapes need approximation.
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | 2(l + w) | l × w |
| Triangle | Sum of three sides | (1/2) × base × perpendicular height |
| Parallelogram | 2(a + b) | base × perpendicular height |
| Trapezium | Sum of four sides | (1/2) × (a + b) × height |
Here r is the radius, d is the diameter and θ is the angle of the sector in degrees. A radius is half the diameter - the most common single error in the whole topic.
| Solid | Volume | Surface area |
|---|---|---|
| Cube | s³ | 6 s² |
| Cuboid | l b h | 2(lb + bh + hl) |
| Cylinder | π r² h | 2 π r(r + h) |
| Sphere | (4/3) π r³ | 4 π r² |
| Cone | (1/3) π r² h | π r(l + r), with l = √(r² + h²) |
A cylinder has radius 7 cm and height 10 cm. Find its volume and its total surface area. Use π = 22/7.
Volume first. Substitute r = 7 and h = 10 into π r² h:
Since 1 litre = 1000 cm³, this cylinder holds 1.54 litres.
Now total surface area, which is two circles plus the curved wall:
Cancel 7 with the denominator: 2 × 22 × 17 = 748 cm².
Check the two parts separately: the two ends are 2 × (22/7) × 49 = 308 cm², and the curved wall is 2 × (22/7) × 7 × 10 = 440 cm². Their sum is 308 + 440 = 748 cm², which matches.
A trapezium has parallel sides of 8 cm and 12 cm and a height of 5 cm. What is its area?
A circle has radius 7 cm. What is its circumference? Use π = 22/7.
What is the volume of a sphere of radius 3 cm?
One cubic metre equals how many litres?
Without looking back: write the area of a triangle and the volume of a cylinder, and state the relationship between one metre and one centimetre before you cube it.
The triangle is half base times height; the cylinder is π r² h; and 1 m = 100 cm, so 1 m³ = 1,000,000 cm³.
The area of a triangle is one half times the base times the . The volume of a cylinder is π r² times . One litre is cm³. The circumference of a circle is 2 π times the .
A cylindrical water tank has radius 0.7 m and height 1.5 m. How many litres does it hold? Use π = 22/7.
Use the cylinder volume formula with r = 0.7 and h = 1.5:
V = π r² h = (22/7) × 0.7² × 1.5 = (22/7) × 0.49 × 1.5.
Since 0.49 ÷ 7 = 0.07, this becomes 22 × 0.07 × 1.5 = 22 × 0.105 = 2.31 m³.
Convert to litres: 1 m³ = 1000 litres, so the tank holds 2310 litres.
The reasoning is the point. Every input was already in metres, so no linear conversion was needed; only the final cubic-to-capacity step was required. A candidate who converted 0.7 m to 70 cm early would get 2,310,000 cm³; the same answer, but far easier to lose a zero in.
Mensuration measures shapes in isolation. The next lesson places them on a grid, where points, distances, midpoints and straight lines are described by coordinates - the bridge from geometry to algebra.