Question 937379: If the difference in the degree measure of interior and exterior angle of a regular polygon is 100, how many sides does it have?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if x is the interior angle, then 180 - x is the exterior angle.
you get:
x + (180 - x) = 180
if the difference between them is equal to 100, then you get:
x - (180 - x) = 100
simplify this to get:
x - 180 + x = 100
combine like terms to get:
2x - 180 = 100
add 180 to both sides to get:
2x = 280
divide both sides by 2 to get:
x = 140.
the formula for the interior angle of a regular polygon is a = 180 * (n-2) / n.
if the interior angle is 140, then the formula becomes:
140 = 180 * (n-2) / n
multiply both sides by n to get:
140 * n = 180 * (n - 2)
simplify to get:
140 * n = 180 * n - 360
add 360 to both sides to get:
140 * n + 360 = 180 * n
subtract 140 * n from both sides to get:
360 = 180 * n - 140 * n
simplify to get:
360 = 40 * n
divide both sides by 40 to get:
360 / 40 = n
solve for n to get:
n = 9.
you can derive the number of sides by use of the external angle as well.
the formula for the number of sides from the external angle is much simpler.
the sum of the external angles of a polygon is always 360, so the number of sides will be 360 / the external angle.
if the internal angle is 140, then the external angle is 180 - 140 = 40.
360 / 40 = 9.
same answer but much simpler to calculate.
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