Numeracy, Maths and Statistics - Academic Skills Kit

Area of a Triangle

Definition

Given the lengths of two sides $a$ and $b$ , and the angle between them $C$ , you can use this formula for the area of a triangle. $\text{area } = \frac{1}{2}ab\sin C\text{.}$

Worked Examples

Example 1

Given $a=10$ , $b=12$ and $C=54^{\circ}$ , find the area of the triangle $ABC$ .

Solution

$\begin{align} \text{Area } &= \frac{1}{2} \times 10 \times 12 \times \sin 54^{\circ}\\ &= \frac{1}{2} \times 120 \times \sin 54^{\circ}\\\\ &= 60\sin 54^{\circ}\\ &= 48.5 \text{ (to 3 sig.fig.)} \end{align}$

Example 2

Given the area of a triangle is $12$ , and the sides $a$ and $b$ are $7$ and $4$ respectively, find the angle between them.

Solution

Label the angle between $a$ and $b$ as $C$ . Now use the area formula:

$\begin{align} 12 &= \frac{1}{2} \times 7\times 4\times \sin C\\ 12 &= \frac{1}{2} \times 28 \times \sin C\\ 12 &= 14 \sin C\\ \sin C &= \frac{6}{7}\\ C &= \sin^{-1}\left(\frac{6}{7}\right)\\ C &= 59^{\circ} \text{ (to 2 sig.fig.)} \end{align}$

Test Yourself

Test yourself: Area of Geometric Shapes