Question 463059: How do you find how many sides would a regular polygon have if you only know each interior angle measures ?
Found 2 solutions by Theo, richard1234:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula for the exterior angle of a polygon is 360 / n.
if you know the interior angle, then 180 - that equals the exterior angle.
you divide the exterior angle into 360 to get the number of sides of the polygon.
for example, we know that a regular pentagon has an interior angle of (n-2)*180 = 3*180 = 540 degrees / 5 = 108 degrees.
suppose we only know that the interior angle is equal to 108 degrees.
we find the exterior angle by subtracting that angle from 180 degrees go get 72 degrees.
we divide 360 by 72 to get 5.
if the angle is not an exact divisor of 360, you will not get a discrete answer, i.e. the number of sides of the polygon will not be an integer.
suppose you are told that the interior angle is 75 degrees.
the exterior angle is equal to 105 degrees.
360 / 105 = 3.428571429 which means the polygon has 3.428571429 sides.
this doesn't make sense, so the polygon can't be a regular polygon (i don't think - at least none of the polygons that i know about have a partial number of sides).
it stands to reason that 360 degrees has to be a multiple of the exterior angle of the polygon in order for the sides to come out as integers.
if somebody else created the polygon using an integral number of sides, then the reverse process described above will get you back to the original number of sides, which will be an integer.
example:
suppose the polygon is 27 sides.
each interior angle of the polygon will measure (25 * 180)/27 = 166.6666666667 degrees (166 + 2/3 degrees).
if that's all you know, and you want to know the number of sides, then divide 360 by (180 - 166.666666667) to get 27.
it works because you are simply reversing the formula that was used to get the angle in the first place.
there are 2 ways to find the interior angle of a polygon.
the first way is to take 360 and divide it by the number of sides and then get the supplement of that.
a polygon with 27 sides will yield an exterior angle of 360 / 27 = 13.3333333 or 13 and 1/3 degrees.
that yields an interior angle of 180 - 13.3333333 which equals 166.666666667 or 166 and 2/3 degrees.
if you know the interior angle, you just reverse the process to get the number of sides.
the other way to get the interior angle of a polygon is to use the formula of (n-2)*180/n
for the 27 sided figure, that yields 25*180/27 = 166.66666667 degrees or 166 and 2/3 degrees.
to reverse the process, you can find the exterior angle and use the process described above, or you can simply reverse engineer the formula for the interior angle by reverse engineering that formula.
the formula for the interior angle of the polygon is (n-2)*180/n = 166.6666667 degrees.
multiply both sides of that formula by n and remove the parentheses to get 180*n - 360 = 166.6666667*n
subtract 166.6666667*n from both sides of that equation and add 360 to both sides of that equation to get 180*n - 166.6666667*n = 360
simplify that equation to get 13.3333333*n = 360
divide both sides of that equation by 13.3333333 to get n = 360 / 13.3333333
simplify that to get n = 27.
If you used the exact number of degrees, then the number of sides should come out as an integer.
If you used a close approximation, then the number of sides should come out to be very close to an integer.
example:
in the example above, if you used 166 and 2/3 degrees for the angle, then the answer should have come out to be exactly 27 sides.
if you used an approximation of 166 and 2/3 degrees, like 166.66667, then the answer should have come out to be very close to 27 sides, but not right on.
If you go out far enough with your approximation, like 166.6666666667, then you might get right on, but no guarantee. It depends on the number of integers you can display on your calculator and the number of digits you carried the fraction out to.
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! Find the sum of the interior angles. This sum is always equal to 180(n-2), where n is the number of sides (this is because you can always draw n-2 non-overlapping triangles in an n-gon). Here, you have 180(n-2) = sum, then solve for n.
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