Area of a triangle with no right angle

Tier: #Higher

🔗What you need to know first

Areas and volumes · Trigonometric functions

How to

For any triangle where you know two sides and the included angle, use the formula: $$\text{Area} = \frac{1}{2}ab\sin C$$

where $a$ and $b$ are two sides and $C$ is the angle between them.

Example: A triangle with sides $a = 7\text{ cm}$, $b = 9\text{ cm}$, and included angle $C = 40°$: $$\text{Area} = \frac{1}{2} \times 7 \times 9 \times \sin 40° = \frac{1}{2} \times 63 \times 0.643 \approx 20.2\text{ cm}^2$$

This formula is essential when the perpendicular height is not given and cannot easily be found. It is derived from the standard $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ using $\text{height} = b\sin C$.

Common error: using the formula with an angle that is not between the two given sides. $C$ must be the angle enclosed by sides $a$ and $b$.

Common topics that go alongside

Substitution Sine rule Cosine rule

❓Questions to practise

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📝Past paper questions

💬What the examiners say

"Many of you used the area formula correctly, which was great. Some of you made basic errors like assuming the triangle was right-angled when it wasn't, or rounded too early and fell outside the range. Show your full working and keep precision until the very end."
"Candidates should remember to write out unrounded answers first as this will normally gain the accuracy mark."
"Centres may wish to emphasise the importance of checking substitutions carefully."

⬆️How you can quickly improve

Check whether the triangle has a right angle — if not, use ½ab sin C where C is the angle between sides a and b.
Identify the included angle (the one between the two known sides) before substituting into the formula.
Keep full calculator precision throughout and only round the final answer.

💡Watch

ℹ️Calculator tricks

Enter $\frac{1}{2} \times a \times b \times \sin(C)$ in one go — use brackets if needed.