A painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure $15.5 \mathrm{~cm}$ and $15.6 \mathrm{~cm}$. She knows that the angle between the fwo sides is $105^{\circ}$. What is the area of the surface she plans to cover with the coating?
The surface area is $\square \mathrm{cm}^{2}$
(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

Step 1 ：The problem provides two sides of a triangle, which are 15.5 cm and 15.6 cm, and the included angle is 105 degrees.

Step 2 ：The area of a triangle given two sides and the included angle can be calculated using the formula: \( Area = \frac{1}{2}ab\sin(C) \) where \( a \) and \( b \) are the sides of the triangle and \( C \) is the included angle.

Step 3 ：In this case, \( a = 15.5 \), \( b = 15.6 \), and \( C = 105 \) degrees. We need to convert the angle from degrees to radians because the sin function uses radians. The conversion is done by multiplying the angle by \( \pi/180 \).

Step 4 ：After converting, we get \( C = 1.8325957145940461 \).

Step 5 ：Substitute the values of \( a \), \( b \), and \( C \) into the formula, we get the area of the triangle is approximately 116.8 square cm.

Step 6 ：Final Answer: The surface area is \(\boxed{116.8} \) \(\mathrm{cm}^{2}\)