Question 1095670: The ratio between an exterior and interior angle is1:8.Find the number of sides
Found 3 solutions by Alan3354, Theo, MathTherapy:
Answer by Alan3354(69443)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the exterior angle and the interior angle are supplementary to each other.
this means that their sum is 180 degrees.
let i = the interior angle and let e equal the exterior angle.
therefore i + e = 180
the ratio between them is 1/8, therefore e/i = 1/8.
solve for e to get e = 1/8 * i
i + e = 180 becomes i + 1/8 * i = 180
multiply both sides of this equation by 8 to get 8i + i = 8 * 180
combine like terms to get 9i = 8 * 180
divide both sides of this equation by 9 to get i = 8 * 20
solve for i to get i = 160 degrees.
this makes e equal to 180 - 160 = 20 degrees.
the number of sides of a polygon can be found by use of the following formula.
n = 360 / e
n = number of sides
e = the exterior angle in degrees.
if you know i, you can find e by using the formula of e = 180 - i.
i = the interior angle in degrees.
we found i and from that found e and from that found the number of sides of the polygon.
in the equation of e/i = 1/8, we could have solve for e directly to get:
i = 8 * e
then i + e = 180 becomes 8 * e + e = 180 which becomes 9 * e = 180 which results in e = 20 degrees.
either way we derived e = 20 degrees and from that were able to derive that the number of sides of the polygon is equal to 360 / e = 360 / 20 = 18.
note that the ratio had to be e/i = 1/8 and could not be i/e = 1/8
if i/e = 1/8, then we solve for e to get e = 8 * i
from that we get i + e = 180 becomes i + 8 * i = 180 which results in 9*i = 180 which results in i = 20.
this makes e = 160 and the number of sides equal to 360 / 160 = 2.25 which can't be because the number of side has to be an integer.
the ration had to be e/i = 1/8.
alternatively i/e = 8 would have worked as well.
both would get you i = 8 * e.
your solution is that the number of sides of the polygon is 18.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website!
The ratio between an exterior and interior angle is1:8.Find the number of sides
With E being an exterior angle, and I, an interior angle, we know that: E + I = 180
Therefore, the measure of an exterior angle is: 
Assuming this is a REGULAR polygon, number of sides is: 
|
|
|
