Question 61248
Here's the question from the book:"An interior angle of a regular polygon is given. Find the number of sides of the polygon.

A regular polygon has all sides the same and also all interior angles are the same.

A regular triangle has 3 internal angles that add up to 180 degrees.
A regular square has 4 internal angles that add up to 360 degrees.
A regular pentagon has 5 internal angles that add up to 540 degrees.

Every time you add a side to a polygon you add 180 degrees.

So, a 3-sided polygon has 1*180 degrees.
A 4-sided polygon has 2*180 degrees.
A 5-sided polygon has 3*180 degrees.

In general, an n-sided polygon has {{{(n-2)*180 degrees}}}.
The interior angle of a regular n-sided polygon, call it I = {{{((n-2)*180)/n}}}.

So, if you know I and want to solve for n, you do the following algebra:

{{{I=((n-2)*180)/n}}}.
Multiply by n: {{{In=(n-2)*180)}}}.
Or, {{{In=180n-360}}}.
Subtract 180n from both sides: {{{In-180n=-360}}}.
Factor out n: {{{n(I-180)=-360}}}
Divide by I-180: {{{n=-360/(I-180)}}}.
Multiply numerator and denominator by -1: {{{n=360/(180-I)}}}.

Verify the equation:

If I = 60 then {{{n=360/(180-60) = 360/120 = 3}}} A triangle!
If I = 90 then {{{n=360/(180-90) = 360/90 = 4}}} A square!
If I = 108 then {{{n=360/(180-108) = 360/72 = 5}}} A pentagon!