Calculate the perimeter and area of triangles, quadrilaterals and circles, and of composite plane figures

What this dot point is asking

SEAB wants you to find the perimeter and area of standard plane figures, triangles, rectangles, parallelograms, trapeziums and circles, and to handle composite shapes made by combining or subtracting these. Reliable use of the standard formulas, with correct units, is the foundation of the mensuration strand.

The answer

Perimeter

The perimeter is the total distance around a shape, found by adding all the outer sides. For a circle the perimeter is called the circumference:

where $r$ is the radius and $d$ the diameter.

Areas of standard shapes

The key area formulas are:

• rectangle: $\text{length} \times \text{width}$,
• triangle: $\dfrac{1}{2} \times \text{base} \times \text{height}$,
• parallelogram: $\text{base} \times \text{height}$,
• trapezium: $\dfrac{1}{2}(a + b)h$, where $a$ and $b$ are the parallel sides,
• circle: $\pi r^2$.

The height in a triangle or parallelogram is the perpendicular height, not a slanted side.

Composite figures

A composite figure is built from standard shapes. Find its area by splitting it into pieces and adding, or by taking a large shape and subtracting a removed piece. The perimeter of a composite shape follows the actual outer boundary, which may include parts of circles.

Units

Area is measured in square units ($\text{cm}^2$, $\text{m}^2$) and perimeter in linear units ($\text{cm}$, $\text{m}$). Convert all lengths to the same unit before calculating, and state the correct unit with the answer.

Examples in context

Example 1. Flooring a room. Tiling an L-shaped room means splitting the floor into two rectangles, finding each area, and adding them to get the total area of tiling needed. Composite-area thinking is exactly what tradespeople do.

Example 2. A running track. The perimeter of a track with two straight sides and two semicircular ends combines straight lengths with the circumference of a full circle (the two semicircles). The total distance is what a runner covers in one lap.

Try this

Q1. Find the area of a triangle with base $12\ \text{cm}$ and perpendicular height $5\ \text{cm}$. [1 mark]

• Cue. $\dfrac{1}{2} \times 12 \times 5 = 30\ \text{cm}^2$.

Q2. Find the circumference of a circle of diameter $10\ \text{cm}$, taking $\pi = 3.142$. [2 marks]

• Cue. $C = \pi d = 3.142 \times 10 = 31.42\ \text{cm}$.

Q3. Find the area of a trapezium with parallel sides $6\ \text{cm}$ and $10\ \text{cm}$ and height $4\ \text{cm}$. [2 marks]

• Cue. $\dfrac{1}{2}(6 + 10) \times 4 = \dfrac{1}{2} \times 16 \times 4 = 32\ \text{cm}^2$.