Q.Find the number of sides of a regular polygon if each of its interior angle is $\dfrac{3\pi}{4}^{c}$.

Since $\pi^{c} = 180°$, an angle of $\theta$ degrees and the same angle of $r$ radians are related by

• To convert degree $\to$ radian, multiply by $\frac{\pi}{180}$.
• To convert radian $\to$ degree, multiply by $\frac{180}{\pi}$.

This single rule is what lets a problem state data in one system (say, a triangle's angles in radian) and ask for the answer in the other, or mix the two freely -- for instance in angle-sum problems for triangles ($180°$), quadrilaterals ($360°$) and regular polygons (exterior angle $=\frac{360°}{n}$, interior angle $=180°-$exterior). Taking $\pi\approx3.14$ (or $\frac{22}{7}$) turns the exact fractional answers into usable decimal or D-M-S approximations, e.g. $1^{c}\approx 57°19'29''$.