To use the Law of Sines to find the length of side \( c \), we first need to understand the Law of Sines formula:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Here, we know \( a = 1246 \) mm and \( \angle A = 72^\circ \). However, we need more information about angles \( B \) and \( C \) to find \( c \). Without those angles or additional sides, we cannot calculate \( c \) directly.
If we assume you have values for angles \( B \) and \( C \) (as \( A + B + C = 180^\circ \)), you could calculate angle \( C \) given \( B \), and then apply the Law of Sines to find side \( c \).
Since I don't have those angles, assume \( \angle B = 90^\circ \) as an example (you can adjust it if you have another angle). This means:
\[ C = 180^\circ - A - B = 180^\circ - 72^\circ - 90^\circ = 18^\circ \]
Now we can use the Law of Sines:
\[ \frac{1246 , \text{mm}}{\sin 72^\circ} = \frac{c}{\sin 18^\circ} \]
Reorganizing the equation to solve for \( c \):
\[ c = \frac{1246 , \text{mm} \cdot \sin 18^\circ}{\sin 72^\circ} \]
Using approximate values for sine:
\[ \sin 72^\circ \approx 0.9511 \] \[ \sin 18^\circ \approx 0.3090 \]
Now substituting these values into the equation:
\[ c \approx \frac{1246 \cdot 0.3090}{0.9511} \approx \frac{385.674}{0.9511} \approx 405.26 , \text{mm} \]
Rounded to the nearest millimeter, \( c \approx 405 , \text{mm} \).
Thus, the answer is 405 mm.