Mensuration is the branch of mathematics concerned with measuring geometric figures — lengths, areas, and volumes. For composite shapes, the strategy is to decompose the figure into standard shapes, compute each component, then add (or subtract).
Addition: when the shape is a combination of non-overlapping standard shapes.
Subtraction: when a region is removed from a larger shape (e.g., a hole, a cutout).
| Shape | Area formula |
|---|---|
| Rectangle | $A = lw$ |
| Triangle | $A = \tfrac{1}{2}bh$ |
| Circle | $A = \pi r^2$ |
| Semicircle | $A = \tfrac{1}{2}\pi r^2$ |
| Trapezium | $A = \tfrac{1}{2}(a+b)h$ |
| Sector | $A = \tfrac{\theta}{360}\pi r^2$ |
A running track consists of a rectangle (100 m × 60 m) with semicircles on each short end.
$$A_{\text{rect}} = 100 \times 60 = 6000 \text{ m}^2$$
$$A_{\text{two semicircles}} = \pi r^2 = \pi (30)^2 = 900\pi \approx 2827 \text{ m}^2$$
$$A_{\text{total}} \approx 6000 + 2827 = 8827 \text{ m}^2$$
Perimeter (track length):
$$P = 2 \times 100 + 2\pi(30) = 200 + 60\pi \approx 388.5 \text{ m}$$
Decompose into prisms, cylinders, pyramids, cones, hemispheres.
A swimming pool is 25 m long, 10 m wide, with a shallow end depth of 1.0 m and deep end depth of 2.2 m.
Model as a trapezoidal prism (trapezium cross-section):
$$V = A_{\text{trapezium}} \times \text{width} = \tfrac{1}{2}(1.0 + 2.2)(25) \times 10 = \tfrac{1}{2}(3.2)(25)(10) = 400 \text{ m}^3$$
Capacity: \$400 \times 1000 = 400{,}000 \text{ L}$.
For surface area: sum the areas of all exposed faces. Internal faces (where parts join) are not included.
Radius $r = 3$ m, cylinder height $h = 5$ m.
$$SA = \underbrace{2\pi r h}{\text{cylinder wall}} + \underbrace{\pi r^2}{\text{base}} + \underbrace{2\pi r^2}_{\text{hemisphere}}$$
$$= 2\pi(3)(5) + \pi(9) + 2\pi(9) = 30\pi + 9\pi + 18\pi = 57\pi \approx 179.1 \text{ m}^2$$
(The top circle of the cylinder is replaced by the hemisphere — do not count it.)
REMEMBER: The circular top of a cylinder that meets a hemisphere is an internal face — it is not part of the surface area. Only external faces contribute.
EXAM TIP: Show the decomposition explicitly: write each component’s formula and calculated value before adding. This earns method marks even if arithmetic errors occur.
