To calculate the area of a non-right-angled triangle, you can use the formula:

In this formula, $a$ and $b$ represent the lengths of two sides of the triangle, while $C$ denotes the measure of the included angle between those two sides. The included angle is the angle that is formed between the two sides you are measuring. The sine function, denoted as $\sin$, is a trigonometric function that can be determined using a scientific calculator or trigonometric tables.

Here is a step-by-step guide on how to apply the formula:

1. Measure or identify the lengths of the two sides of the triangle, denoting them as $a$ and $b$.
2. Measure or identify the included angle $C$ between these two sides.
3. Use a calculator to compute the sine of the included angle $\sin(C)$.
4. Multiply the lengths of the two sides together: $a \times b$.
5. Multiply the result by $\sin(C)$.
6. Finally, multiply this value by $\frac{1}{2}$ to obtain the area.

For example, 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. First, find $\sin(60^\circ)$, which is approximately $0.866$.
2. Next, multiply the lengths of the sides: $$5 \times 7 = 35$$
3. Then, multiply this result by the sine of the angle: $$35 \times 0.866 \approx 30.31$$
4. Finally, multiply by $\frac{1}{2}$: $$\frac{1}{2} \times 30.31 \approx 15.155$$

Thus, the area of the triangle is approximately $15.155$ square centimeters.