How many sides does a regular polygon have if each interior angle is three times the measure of each exterior angle measure?
Let the measure of each exterior angle be x.
By definition, the sum of the measures of the interior angles of a polygon with n sides is given by (n-2) * 180 degrees.
Since each interior angle is three times the measure of the corresponding exterior angle, the measure of each interior angle is 3x.
Therefore, we have the equation (n-2) * 180 = n * 3x.
Expanding this equation gives 180n - 360 = 3nx.
Rearranging this equation, we get 180n - 3nx = 360.
Factoring out n, we obtain n(180 - 3x) = 360.
Dividing both sides of the equation by (180 - 3x) gives n = 360 / (180 - 3x).
Since a polygon must have a whole number of sides, n must be an integer.
We can find the values of x for which n is an integer by finding the factors of 360 that make (180 - 3x) a factor as well.
The factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
Checking each of these values for x, we find that x = 60 is the only factor that makes (180 - 3x) also a factor.
Substituting x = 60 into the equation n = 360 / (180 - 3x), we get n = 360 / (180 - 3*60) = 5.
Therefore, a regular polygon with each interior angle being three times the measure of each exterior angle has 5 sides.