To find the area of a triangle when two angles and one side are given, we can use the formula based on the sine rule:

$\text{Area} = \frac{1}{2} c^2 \times \frac{\sin(A) \sin(B)}{\sin(C)}$

We are given:

• $A = 40.8^\circ$
• $B = 39.6^\circ$
• $c = 27.1 \, \text{m}$

First, we need to find the third angle $C$, since the sum of the angles in a triangle is $180^\circ$:

$C = 180^\circ - (A + B) = 180^\circ - (40.8^\circ + 39.6^\circ) = 99.6^\circ$

Now, use the formula for the area:

$\text{Area} = \frac{1}{2} \times c^2 \times \frac{\sin(A) \sin(B)}{\sin(C)}$

Substituting the values:

$\text{Area} = \frac{1}{2} \times (27.1)^2 \times \frac{\sin(40.8^\circ) \times \sin(39.6^\circ)}{\sin(99.6^\circ)}$

Now, calculate the sines of the angles and the area. I will compute this for you.The area of the triangle is approximately $155.1 \, \text{m}^2$, rounded to the nearest tenth.

Do you have any further questions or would you like more details on this solution?

Here are 5 related questions to explore:

1. How can you find the third side of the triangle using the Law of Cosines?
2. What would the area be if side $c$ was longer?
3. How does the sine rule apply when the triangle is right-angled?
4. Can you find the height of the triangle given the same information?
5. How would the area change if angle $C$ was obtuse?

Tip: Always make sure to convert angles into radians when using trigonometric functions in most programming environments.