why when we add more side, the sum of the angles go up by 180 degrees

Question

The sum of the angles of a triangle is 180

For a square it’s 360

For a pentagon it’s 540

For a hexagon it’s 720

Why is that?

In: Mathematics

Answer 1

Take any corner of a shape.

Imagine travelling down one of the sides to that corner (and for ease imagine going clockwise).

Rather than thinking about the angle inside the shape, think of instead of the angle you had to turn at the corner to continue to follow the shape.

That is 180° minus the angle inside the shape.

But once you turn every corner of the shape, you’re back going in the same direction so the sum of all those turns is 360°.

But the sum of all those turns is also the number of corners (which is also the number of sides) × 180° minus the sum of all the angles inside the shape.

So 180° × N – sum of angles inside shape = 360°. And so the sum of angles inside a N-gon is 180° × (N – 2).