Final answer:
To find the number of sides of a regular polygon where an interior angle is five times an exterior angle, set up an equation based on the knowledge that the sum of an interior and exterior angle is 180 degrees. Solve for the exterior angle, then divide 360 by the measure of the exterior angle to find that the polygon has 12 sides.
Step-by-step explanation:
To calculate the number of sides a regular polygon has where an interior angle is five times the size of its exterior angle, we start by understanding the relationship between interior and exterior angles of a polygon. For any polygon, the sum of an interior angle and its adjacent exterior angle is 180 degrees because they form a straight line. Since we know an interior angle (I) is five times an exterior angle (E), we can express this as I = 5E. Using the relationship I + E = 180, we can substitute for I:
5E + E = 1806E = 180E = 30
Now that we have the measure of an exterior angle, we can find the number of sides (n) by using the fact that the sum of all exterior angles of a polygon is 360 degrees. The formula is:
n = 360 / En = 360 / 30n = 12
Therefore, a regular polygon where an interior angle is five times the size of its exterior angle has 12 sides.