Area of a Triangle

Video lesson4 worked examples

Theory

The area of any non-right-angled triangle can be calculated if you know the lengths of two sides and the size of the angle included between them.

The formula provided on the exam sheet is:

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

Candidates may also be required to work backwards from a given area to calculate a missing angle or side.

The Golden Rule: the angle must be the one between the two sides you use (the included angle). Working backwards, substitute the known values and rearrange for $\sin C$ — and if the angle is described as obtuse, take $180^\circ$ minus the calculator value.

⚠️ Common Examiner Traps

Wrong angle: the formula needs the angle between the two sides — not just any angle in the triangle.
The obtuse case: $\sin C = \sin(180^\circ - C)$ , so a value like $\sin C = 0.5$ gives both $30^\circ$ and $150^\circ$ — pick the one the question asks for.
Regular polygons: split them into identical triangles from the centre and add the areas.
Rounding: keep full accuracy in $\sin$ until the final line.

Worked examples

Example 1

A triangle has sides of 8 cm and 12 cm with an included angle of 30°. Calculate its area.

Step 1: Substitute into the formula:

A = \frac{1}{2} \times 8 \times 12 \times \sin(30^\circ)

Step 2: Since $\sin(30^\circ) = 0.5$ :

A = 4 \times 12 \times 0.5 = 24\text{ cm}^2

Example 2

A triangle with sides 10 m and 6 m has an area of 15 m². Find the included angle, $C$ .

Step 1: Set up the equation:

15 = \frac{1}{2} \times 10 \times 6 \times \sin C

Step 2: Simplify:

15 = 30 \sin C

Step 3: Divide by 30:

\sin C = 0.5

Answer: $C = \sin^{-1}(0.5) = 30^\circ$ .

Example 3

A triangle has an area of 40 cm², and sides of 10 cm and 16 cm. Calculate the size of the obtuse included angle.

Step 1: Set up the equation:

40 = \frac{1}{2} \times 10 \times 16 \times \sin C

Step 2: Simplify:

40 = 80 \sin C

Step 3: Solve for $\sin C$ :

\sin C = 0.5

Since the angle is obtuse, use the CAST diagram (Quadrant 2):

Answer: $C = 180^\circ - 30^\circ = 150^\circ$ .

Example 4

🔗 Bringing it together (regular polygon)

Calculate the area of a regular hexagon with side length 8 cm, giving your answer as an exact surd.

Step 1: A regular hexagon splits into 6 identical triangles from the centre. Each has two sides of 8 cm (the radius equals the side for a hexagon) and an included angle of $360^\circ \div 6 = 60^\circ$ .

Step 2: Area of one triangle, using $\sin 60^\circ = \frac{\sqrt{3}}{2}$ :

\frac{1}{2} \times 8 \times 8 \times \frac{\sqrt{3}}{2} = 16\sqrt{3}

Step 3: Multiply by the 6 triangles:

6 \times 16\sqrt{3} = 96\sqrt{3}

Answer: $96\sqrt{3} \approx 166.3$ cm².

Ready to practise?

11 Triangle Area questions with answers and video solutions