Perimeter, area and volume are the three core measurement calculations in Foundation Mathematics. They appear constantly in real-world contexts: fencing a paddock, tiling a floor, filling a fish tank. The key is selecting the correct formula for the shape and applying it carefully.
KEY TAKEAWAY: Perimeter is a length (one dimension), area is a surface (two dimensions), and volume is a space (three dimensions). Each has different units: m, m², m³.
Perimeter is the total distance around the outside of a shape.
$$P = \text{sum of all side lengths}$$
| Shape | Formula |
|---|---|
| Rectangle | $P = 2(l + w)$ |
| Square | $P = 4s$ |
| Triangle | $P = a + b + c$ |
| Circle (circumference) | $C = 2\pi r = \pi d$ |
Worked Example:
A rectangular garden is $8.5\text{ m}$ long and $3.2\text{ m}$ wide. Find the perimeter.
$$P = 2(8.5 + 3.2) = 2 \times 11.7 = 23.4\text{ m}$$EXAM TIP: Circumference uses $\pi \approx 3.14159$. Unless told otherwise, use the $\pi$ button on your calculator for maximum accuracy.
| Shape | Formula | Variables |
|---|---|---|
| Rectangle | $A = lw$ | $l$ = length, $w$ = width |
| Square | $A = s^2$ | $s$ = side length |
| Triangle | $A = \frac{1}{2}bh$ | $b$ = base, $h$ = perpendicular height |
| Circle | $A = \pi r^2$ | $r$ = radius |
| Parallelogram | $A = bh$ | $b$ = base, $h$ = perpendicular height |
| Trapezium | $A = \frac{1}{2}(a+b)h$ | $a, b$ = parallel sides, $h$ = height |
Worked Example — Triangle:
A triangular vegetable patch has base $6\text{ m}$ and perpendicular height $4\text{ m}$.
$$A = \frac{1}{2} \times 6 \times 4 = 12\text{ m}^2$$
Worked Example — Circle:
A circular fountain has radius $2.5\text{ m}$.
$$A = \pi \times (2.5)^2 = \pi \times 6.25 \approx 19.6\text{ m}^2$$COMMON MISTAKE: Using the slant height of a triangle instead of the perpendicular height. The height $h$ must be measured at a right angle to the base.
Real objects often combine simple shapes. Break them into parts, calculate each area separately, then add or subtract.
Worked Example:
An L-shaped floor: overall rectangle $10\text{ m} \times 6\text{ m}$, with a $3\text{ m} \times 4\text{ m}$ rectangle cut out.
$$A = (10 \times 6) - (3 \times 4) = 60 - 12 = 48\text{ m}^2$$
General principle for prisms and cylinders:
$$V = \text{Area of cross-section} \times \text{length (or height)}$$
| Object | Formula | Variables |
|---|---|---|
| Rectangular prism | $V = lwh$ | $l, w, h$ = length, width, height |
| Cube | $V = s^3$ | $s$ = side length |
| Cylinder | $V = \pi r^2 h$ | $r$ = radius, $h$ = height |
| Triangular prism | $V = \frac{1}{2}bhl$ | $b, h$ = triangle base/height, $l$ = length |
Worked Example — Cylinder:
A cylindrical water tank has radius $1.2\text{ m}$ and height $2.0\text{ m}$.
$$V = \pi \times (1.2)^2 \times 2.0 = \pi \times 1.44 \times 2.0 \approx 9.05\text{ m}^3$$
Capacity:
$$9.05\text{ m}^3 = 9.05 \times 1000 = 9050\text{ L}$$
Worked Example — Rectangular Prism:
A concrete slab is $4\text{ m}$ long, $3\text{ m}$ wide, $0.1\text{ m}$ thick.
$$V = 4 \times 3 \times 0.1 = 1.2\text{ m}^3$$
| Measurement | Unit Examples |
|---|---|
| Perimeter | mm, cm, m, km |
| Area | mm², cm², m², ha, km² |
| Volume / Capacity | mm³, cm³, m³, mL, L, kL |
VCAA FOCUS: Multi-step problems are common — for example, find area of a shape, convert units, then calculate cost (e.g. at $\$45$ per m²). Practise the full chain: formula → calculate → convert → apply rate.
