SOLUTION: please help me solve this question: each interior angle of a regular polygon is four times the measure of each exterior angle. how many diagonals does the polygon have?

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Question 478149: please help me solve this question: each interior angle of a regular polygon is four times the measure of each exterior angle. how many diagonals does the polygon have?

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
each interior angle is equal to 4 times the exterior angle.
ia = interior angle
ea = exterior angle
ie ea = x then ia = 4x
now the interior angle of a polygon and its exterior angle are supplementary.
this means that ia + ea = 180
this means that 4x + x = 180 which means that 5x = 180 which means that x = 36 degrees which means that 4x = 144 degrees.
ea = 36 degrees
ia = 144 degrees
now the sum of the exterior angles of a polygon is always equal to 360 degrees.
if you divide 360 by 36 you get 10.
this means that the polygon has 10 sides.
it's a decagon.
so far so good.
now you want to determine how many diagonals the decagon has.
the formula for determining the number of diagonals in a polygon is:
d = n(n-3)/2
d is the number of diagonals
n is the number of sides.
if it's a triangle, then the number of diagonals is equal to 3*0/2 = 0
if it's a quadrilateral, then the number of diagonals is 4*1/2 = 2
pentagon = 5*2/2 = 5
hexagon = 6*3/2 = 9
septagon = 7*4/2 = 14
octagon = 8*5/2 = 20
nonagon or enneagon (9 sides) = 9*6/2 = 27
decagon = 10*7/2 = 35
the decagon will have 35 diagonals.