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.
| 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.
