Area of a Triangle (SQA National 5 Maths): Revision Note

Swapping the letters around allows the triangle area formula to be written in these forms as well:

$\mathrm{Area}=\frac{1}{2}ac\sin B$

$\mathrm{Area}=\frac{1}{2}bc\sin A$

Just make sure the angle used in the formula is in between the two sides used.

The area of a triangle formula appears on the Formulae List in the exam paper in the form

$A=\frac{1}{2}ab\sin C$

Remember that the $A$ there is for 'Area'. It is not referring to an angle labelled $A$!

Triangle $\mathrm{ABC}$ is shown in the diagram.

• Angle ABC = 109°

• AB = 8.1 cm

• BC = 12.3 cm

Calculate the area of the triangle.

Answer:

Label the sides of the triangle

You want to find the area and are given the size of angle B

• So swap the letters around in the Formulae List form of the triangle area formula to get $\mathrm{Area}=\frac{1}{2}ac\sin B$

Substitute the known values into that equation

$\begin{array}{rcl}\mathrm{Area}& =& \frac{1}{2}\times 8.1\times 12.3\times \sin \left(109\right)\\ & =& 47.101007...\end{array}$

Round to a sensible degree of accuracy

• Unless a question tells you otherwise, 3 significant figures is usually a good choice

47.1 cm²  (3 s.f.)

In the diagram

• Angle BAC = 26°

• AC = 30 cm

The area of triangle ABC is 160 square centimetres.

Calculate the length of AB.

Answer:

Label the sides of the triangle

You want to find the length of c, and you know the area, the length of b and the angle at A

• So swap the letters around in the Formulae List form of the triangle area formula to get $\mathrm{Area}=\frac{1}{2}bc\sin A$

Substitute the known values into that equation

$\begin{array}{rcl}160& =& \frac{1}{2}\times 30\times c\times \sin \left(26\right)\end{array}$

Rearrange to make c the subject

$\begin{array}{rcl}c& =& \frac{\left(2\times 160\right)}{30\times \sin \left(26\right)}\\ & =& 24.332501...\end{array}$

Round to a sensible degree of accuracy

• Unless a question tells you otherwise, 3 significant figures is usually a good choice

24.3 cm  (3 s.f.)