Question

The formula of each interior angle of an n-sided regular polygon is:

Answer 1

Hint: Working our way upwards from $n = 3,$ we will manually find the interior angles for a few polygons and try to establish a relationship for a general n sided regular polygon.

Complete step-by-step answer:
(square) For $n = 4,$ it’s $90^\circ $
(regular pentagon) For $n = 5,$ it’s $108^\circ $
(regular hexagon) For$n = 6,$it’s $120^\circ $

Right off the bat, we can’t seem to find a direct relation so let's find the SUM of all interior angles and try to relate it to n.

n $\sum $interior angles
3 180
4 360
5 540
6 720

Here, the pattern is more clear.
$\sum $Interior angles \[ = 180\left( {n - 2} \right)\]
But because these are regular polygons, each interior ${\text{angle}} = \dfrac{{\sum {\text{interior}}\;{\text{angles}}}}{n}$

Interior angle is given by $ = \boxed{180\left( {\dfrac{{n - 2}}{n}} \right) = 180\left( {\dfrac{{n- 2}}{n}} \right)}$

Note: You can also directly try to relate interior angles to n by trial and error but it may take more time. In the end, we need to arrive at some data that is easy to extrapolate, and generalize, so proceed towards this goal.