20.8 The area of any triangle — Learn

Prerequisites: triangle area, sine ratios, and matching sides to angles.

Objectives: use two sides and their included angle to find a triangle's area, or rearrange the formula to find a missing side or angle.

When perpendicular height is unknown, use $\text{Area}=\frac12 ab\sin C$. The angle $C$ must lie between the known sides $a$ and $b$.

Example 1. Find the area when two sides are $8$ cm and $5$ cm, with included angle $30^\circ$.

Example 2. A triangle has sides $7$ cm and $9$ cm with included angle $40^\circ$. Find its area to 3 significant figures.

Example 3. A triangle has area $20$ cm$^2$ and sides $8$ cm and $10$ cm. Find its acute included angle $C$.

The other angle with sine $\frac12$ is $150^\circ$. The question asks for the acute angle, so $30^\circ$ is the answer.

Example 4. A triangle has area $20$ cm$^2$, one side is $10$ cm, and the included angle is $30^\circ$. Find the other side $b$.

Watch out. The angle must be between the two given sides. If it is opposite one of them, mark the included angle before using the formula.

Progress check

For the garden rectangle, use base times perpendicular height for its area. For a slanted triangle, the sine factor supplies the perpendicular height.

You should now be able to

• use $\frac12 ab\sin C$ to find the area from two sides and their included angle;
• rearrange the formula to find a missing side or angle;
• check that the angle is included and give the correct units.