What does the 180 in the formula (n-2)180 represent?

Question

this is the formula for the sum of the interior angles of an n gon

(n-2)180

what does the 180 in the formula though show?

Answer 1

angle

Answer 2

Since the least total value of interior angles you can have is 180, which is within the triangle, it is just taking the least possible total interior angle value and multipling it by the number of sides minus 2.

Let me show you what i mean

Triangle
(3 - 2) * 180 = 1 * 180 = 180 total measure for the interior angles

Quadrilateral
(4 - 2) * 180 = 2 * 180 = 360 or 2 triangles put together

Pentagon
(5 - 2) * 180 = 3 * 180 = 540 or 3 triangle put together

and so on.

So in short, its the total sum of the interior angles of a triangle times the least number of triangles it would take to make up that shape.

Answer 3

you might actually be looking for the answer "sum of angles in a triangle"

To convince yourself of this, try a few...

Draw a rectangle, and a line across a diagnal. This gives you 2 triangles, and (4-2)(180) = 360 degrees total.

For a pentagon, you'd pick a point and connect it to two opposite points, giving you 3 triangles, or (5-2)(180) = 540 degrees total.

So for your n-sided-gons, you're actually adding more sides by adding more triangles.

Answer 4

Degrees. 180 degrees is a straight angle.

Answer 5

you get it from the derivation.

for an n-agon
each angle "bends" 180°-θ adding them up must give 360°

n*180°-sum of θ's=360°
sum of angles=n*180°-360°=n*180-2*180
sum of θ=180°(n-2)