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To solve this problem, we need to use the formula for the area of a triangle when two sides and the included angle are given. The formula is:

\[
A = \frac{1}{2} \times a \times b \times \sin(C)
\]

Where:
- \(a = 13.1 \, \text{cm}\)
- \(b = 18.4 \, \text{cm}\)
- \(C = 120^\circ\)

Now, let's compute the area step by step.

1. First, calculate \( \sin(120^\circ) \). Since \(\sin(120^\circ) = \sin(60^\circ)\), we know:

\[
\sin(120^\circ) = \frac{\sqrt{3}}{2} \approx 0.866
\]

2. Now, substitute the values into the formula:

\[
A = \frac{1}{2} \times 13.1 \times 18.4 \times 0.866
\]

3. First, calculate \( 13.1 \times 18.4 = 241.04 \, \text{cm}^2 \).

4. Then, multiply that by \(0.866\):

\[
241.04 \times 0.866 \approx 208.77 \, \text{cm}^2
\]

5. Finally, divide by 2 to get the area:

\[
A = \frac{208.77}{2} \approx 104.4 \, \text{cm}^2
\]

Thus, the area of the surface she plans to cover is approximately **104.4 cm²**.

Do you want any further details or have any questions about this? 

Here are some related questions:
1. How do you compute the sine of an angle using a calculator?
2. What happens if the angle between the sides was different, say \(90^\circ\)?
3. How can you find the area of a triangle if only all three sides are given?
4. What is the significance of the included angle in finding the area of a triangle?
5. How does this formula relate to Heron's formula for the area of a triangle?

**Tip:** When solving triangle problems involving angles and sides, always check if using trigonometric functions is necessary, as it often simplifies calculations.

A painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure 13.1 cm and 18.4 cm. She knows that the angle between the two sides is 120°. What is the area of the surface she plans to cover with the coating?

Learn how to find the area of a triangle with sides 13.1 cm and 18.4 cm, and an included angle of 120°. This trigonometry problem shows step-by-step calculations using the formula A = (1/2) * a * b * sin(C). Suitable for high school students.

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