To find the measure of angle \( C \) in triangle \( ABC \) given the sides \( a \), \( b \), and \( c \), we can use the Law of Cosines, which states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Where:
- \( a = 46 \) m
- \( b = 30 \) m
- \( c = 20 \) m
- \( C \) is the angle opposite side \( c \).
First, we rearrange the formula to solve for \( \cos(C) \):
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]
Now, we substitute the given values into the formula:
\[ a^2 = 46^2 = 2116 \] \[ b^2 = 30^2 = 900 \] \[ c^2 = 20^2 = 400 \]
Now, plug these into the formula for \( \cos(C) \):
\[ \cos(C) = \frac{2116 + 900 - 400}{2 \times 46 \times 30} \]
Calculating the numerator:
\[ 2116 + 900 - 400 = 2616 \]
Calculating the denominator:
\[ 2 \times 46 \times 30 = 2760 \]
Now, we find \( \cos(C) \):
\[ \cos(C) = \frac{2616}{2760} \]
Now, we simplify:
\[ \cos(C) \approx 0.948 \]
Now we find angle \( C \) by taking the inverse cosine:
\[ C = \cos^{-1}(0.948) \]
Using a calculator:
\[ C \approx 18.19^\circ \]
Rounding to the nearest degree:
\[ C \approx 18^\circ \]
Thus, the measure of \( \angle C \) is approximately \( \boxed{18}^\circ \).