Area of a Triangle

Syllabus

determine the area of any triangle, given two sides and an included angle, by using the rule $A=\frac{1}{2}ab\sin C$, and solve related practical problems

For non-right angled trigonometry, we label the triangles as shown below:

Important

Sides are denoted with a lowercase letter and angles uppercase. $\angle A$ is opposite $a$, $\angle B$ is opposite $b$ and $\angle C$ is opposite $c$.

note

A labelled triangle is not on your reference sheet.

Deriving The Formula (Extension)

In stages 4 and 5.2, to calculate the area of a triangle you would have used the formula $A=\frac{1}{2}bh$ where $b$ is the length of the base and $h$ is the perpendicular height.

Using right-angled trigonometry we can express $h$ in terms of the side lengths and angles of the triangle.

$$\begin{aligned}\sin C&=\frac{\text{opp}}{\text{hyp}}\\\sin C&=\frac{h}{a}\\ a\sin C&=h\\h&=a\sin C\end{aligned}$$

Which can be substituted into $A=\frac{1}{2}bh$ to give $A=\frac{1}{2}ab\sin C$

Applying The Formula

Important

$\text{Area}=\frac{1}{2}ab\sin C$

note

This is reference sheet

note

Don't forget that $A=\frac{1}{2}bh$ is still a valid way of calculating the area of a triangle.

Example 1

Calculate the area of the triangle below. Answer correct to 3 significant figures.

Solution

$$\begin{aligned} A&=\frac{1}{2}ab\sin C\\ A&=\frac{1}{2}(8.2)(10.7)\sin 32^\circ&&\text{(Substitute)}\\ A&=23.247...\\ A&\approx23.2\text{ cm}^2&&\text{(Rounded to appropraite number}\\ &&&\text{of s.f. and include units)} \end{aligned}$$

Example 2

Calculate the area of the triangle below. Answer correct to 3 significant figures.

Solution

Since the two provided sides are the same, the triangle must be isosceles. This means that both of the base angles must be $70^\circ$.

The required angle must therefore be $40^\circ$ (all angles in a triangle sum to $180^\circ$).

$$\begin{aligned} A&=\frac{1}{2}ab\sin C\\ A&=\frac{1}{2}(9)(9)\sin 40^\circ&&\text{(Substitute)}\\ A&=26.032...\\ A&\approx26.0\text{ cm}^2&&\text{(Rounded to appropraite number}\\ &&&\text{of s.f. and include units)} \end{aligned}$$