There are a couple of ways to understand the reasoning behind the formula. First, is the “360° turn” method. Constructing the sides of a polygon is like walking and making a bunch of angled turns until you reach the start again. The totality of those angles puts you back in your original orientation and thus must add up to 360°.

So, using an equilateral triangle as the simplest example, imagine you start at the triangle’s left base vertex. You then start walking east to the right vertex and make your first turn to face the top vertex. The turn you would make is actually 120° away from your original heading, and that’s what forms the 60° on the interior of the triangle. You then walk some more to the top, make your second 120° turn, then walk to your start and make another 120° turn so you face your original orientation. Together, the three 120° turns equal 360°, which is

[math]120 + 120 + 120 = 360 [/math]

You can write this equation in an equivalent form that references the internal angles of the triangle instead of the 120° turns you made:

[math](180 - a_1) + (180 - a_2 ) + (180 - a_3 ) = 360[/math]

Note that because you made 3 turns for an n-sided polygon, there are also three (180-a) terms.

Rewriting the same equation, you get:

[math](3 * 180) - ( a_1 + a_2 + a_3) = (2 * 180)[/math]

Rewriting again:

[math](3 * 180) - (2 * 180) = (a_1 + a_2 + a_3)[/math]

And again:

[math](3 - 2) * 180 = (the sum of the internal angles)[/math]

Now note that the “3” arises from the number of turns you made, i.e. the n-sides of the polygon. So finally, we get:

[math](n - 2) * 180 = the sum of the internal angles[/math]

This equation applies to any n-side polygon because as you make more turns, it’s simply adding more (180 - a) terms to the summation that should still equal 360.

The second method for understanding (n-2)*180 is by slicing any n-side polygon into composite triangles by drawing lines between vertexes. For a triangle, no slicing is needed (because it’s already a triangle). And as we know, the sum of the internal angles of a triangle is 180.

Looking at a 4-sided polygon (a rectangle), you end up with two composite triangles. The internal angles of those two triangles will all add up to the total internal angles of the 4-sided polygon. So:

[math]2 * 180 = sum of internal angles[/math]

Slicing a 5-sided pentagon, you get 3 composite triangles, and so:

[math]3 * 180 = sum of internal angles[/math]

It should be somewhat obvious that the sum of the internal angles is thus dependent on how many composite triangles the n-sided polygon can be sliced into. And that number of triangles is given by:

[math](n - 2)[/math]

So if (n-2) triangles are present within an n-sided polygon, then the sum of the internal angles of those composite triangles is

[math](n - 2) * 180 = sum of internal angles[/math]