Area Formulae: Triangles, Parallelograms and Trapezia

Key Area Formulae

Critical: The height is always the perpendicular (right-angle) height — not the slant side.

Triangle Area

$$A_{\triangle} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}$$

Example: A triangle has base 12 cm and perpendicular height 7 cm. $$A = \frac{1}{2} \times 12 \times 7 = 42 \text{ cm}^2$$

Right-angled triangle: The two shorter sides are the base and height. $$A = \frac{1}{2} \times a \times b$$

Rearranging: If you know the area and base, find the height: $h = \dfrac{2A}{b}$.

Parallelogram Area

$$A_{\text{parallelogram}} = b \times h$$

The height must be perpendicular to the base (not the slant height).

Example: Parallelogram with base 9 cm, slant side 11 cm, perpendicular height 6 cm. $$A = 9 \times 6 = 54 \text{ cm}^2 \quad (\text{NOT } 9 \times 11)$$

Trapezium Area

$$A_{\text{trapezium}} = \frac{1}{2}(a + b) \times h$$

where $a$ and $b$ are the two parallel sides and $h$ is the perpendicular height between them.

Example: Trapezium with parallel sides 5 cm and 11 cm, height 4 cm. $$A = \frac{1}{2}(5 + 11) \times 4 = \frac{1}{2} \times 16 \times 4 = 32 \text{ cm}^2$$

Composite Shapes

Break compound shapes into standard shapes, calculate each area and add (or subtract).

Example: An L-shape can be split into two rectangles. Calculate each rectangle's area and sum them.

Area Units

• $1 \text{ m}^2 = 10{,}000 \text{ cm}^2$
• $1 \text{ cm}^2 = 100 \text{ mm}^2$
• $1 \text{ km}^2 = 1{,}000{,}000 \text{ m}^2$

WJEC Exam Tips

• Always identify the perpendicular height — it may not be labelled, requiring Pythagoras to find it.
• Units: check the units are consistent (e.g. mix of mm and cm) before calculating.
• Trapezium formula is given on the WJEC formula sheet at Foundation/Intermediate but not always at Higher — learn it.
• For composite shapes: show how you split the shape and the area of each part.