To determine the area of a triangle when you know the lengths of two sides and the included angle, you can use the formula:

where $a$ and $b$ are the lengths of the two sides, and $C$ is the measure of the included angle.

This formula is particularly advantageous when you do not have the height of the triangle but do have the lengths of two sides and the angle formed between them. Here’s a step-by-step guide on how to apply this formula:

1. Identify the lengths of the two sides, denoted as $a$ and $b$.
2. Measure or determine the included angle $C$ in either degrees or radians.
3. Use a scientific calculator to compute $\sin(C)$. Ensure that your calculator is set to the correct mode: degrees if $C$ is in degrees, and radians if $C$ is in radians.
4. Multiply the lengths of the two sides: $a \times b$.
5. Multiply this result by the sine of the included angle: $a \times b \times \sin(C)$.
6. Finally, multiply by $\frac{1}{2}$ to obtain the area:

Example Calculation

Consider a triangle with sides $a = 5$ cm, $b = 7$ cm, and the included angle $C = 60^\circ$. The area can be calculated as follows:

1. Calculate $\sin(60^\circ)$:

2. Multiply the lengths of the sides:

3. Multiply this result by $\sin(60^\circ)$:

4. Finally, multiply by $\frac{1}{2}$:

Thus, the area of the triangle is approximately $15.16 \, \text{cm}^2$.