At each vertex of any polygon there is an interior angle and an exterior angle. The acute* angle between the two sides is the exterior one, and the obtuse angle is the interior one.
They add up to 180.
So for an n sided polygon the total of all these angles is 180n.
If you imagine a little ant walking around the perimeter of the polygon, each time it reaches a vertex it has to turn through the acute angle to carry on the walk along the next side. After it has completed the walk it has turned a total of 360. So the sum of the externals is 360 (true for all polygons).
Therefore the sum of the internals is 180n - 360 = 180(n-2)
eg. For a triangle, n = 3 , so the internals add to 180(3-2) = 180
For a quadrilateral, n = 4, so sum = 180(4-2) = 360
hexagon, n = 6, sum = 180(6-2) = 180 x 4 = 720.
and so on.
Bob
* If the polygon has 5 or more sides. For n = 3 and 4, reverse acute and obtuse.
]]>Therefore n=15
]]>