Question 1072176: Two regular polygons are such that the ratio between their number of sides is 1:3 and the ratio of measures of their interior angles is 3:4. Find the number of sides of each polygon.
Found 2 solutions by htmentor, Boreal:
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! Ratio of sides, n/m = 1/3 -> m = 3n
Ratio of angles, N/M = 3/4 -> M = 4/3N
The interior angle of a regular polygon with n sides is (n-2)*180/n
So we have N = (n-2)*180/n and M = 4/3N = (3n-2)*180/3n
We have two equations and two unknowns.
We can eliminate N and solve for n, the number of sides:
(n-2)*180/n = (3n-2)*180/4n
n-2 = (3n-2)/4
This gives n = 6
Thus m = 3*6 = 18
So the number of sides are 6 and 18.
Answer by Boreal(15235)  (Show Source):
You can put this solution on YOUR website! There is likely a far easier way to do this, but this way gets there.
the number of sides*the interior angle is a 4:1 ratio, (3/1)(4/3)
The sum of the interior angles for n1 is n1(180n1-360)/n1 divided by
n2(180 n2-360)n2
this ratio is 4.
cross-multiply and remove the leading n1 and n2 by canceling with the denominator
4(180n1-360)=180n2-360
but n2=3n1
720n1-1440=540n1-360
180 n1=1080
n1=6, hexagon, with interior angles 120 degrees
n2=18, with interior angles 160 degrees, 4:3 ratio.
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