In a given regular polygon, the ratio of the exterior angle to the interior angles is 1:3. How many side has the polygon?

Answer Details

In any polygon, the sum of the exterior angles is always 360 degrees. Therefore, the measure of each exterior angle of a regular polygon with n sides is 360/n degrees. The ratio of the exterior angle to the interior angle is 1:3. This means that the measure of the exterior angle is three times the measure of the interior angle. Let x be the measure of the interior angle. Then, the measure of the exterior angle is 3x. We know that the sum of the interior angles of any polygon is (n-2) times 180 degrees. Therefore, the measure of each interior angle of a regular polygon with n sides is [(n-2) x 180]/n degrees. Now we can set up an equation to solve for n. 3x = 360/n x = (n-2) x 180/n Substituting 3x for the exterior angle in the first equation: 3x = 360/n x = 120/n Substituting 120/n for x in the second equation: 120/n = (n-2) x 180/n Simplifying: 120 = 180(n-2) 120 = 180n - 360 540 = 180n n = 3 However, a polygon with 3 sides is not possible since it would be a triangle. Therefore, the answer is not 3. We can try other answer options by substituting each value of n in the formula for the measure of each exterior angle (360/n) and checking if the ratio of the exterior angle to the interior angle is 1:3. For n=4, the measure of each exterior angle is 90 degrees, and the measure of each interior angle is 90 degrees. Therefore, the ratio of the exterior angle to the interior angle is 1:1, which is not 1:3. For n=5, the measure of each exterior angle is 72 degrees, and the measure of each interior angle is 108 degrees. Therefore, the ratio of the exterior angle to the interior angle is 2:3, which is not 1:3. For n=6, the measure of each exterior angle is 60 degrees, and the measure of each interior angle is 120 degrees. Therefore, the ratio of the exterior angle to the interior angle is 1:2, which is not 1:3. For n=8, the measure of each exterior angle is 45 degrees, and the measure of each interior angle is 135 degrees. Therefore, the ratio of the exterior angle to the interior angle is 1:3, which satisfies the given condition. Therefore, the polygon has 8 sides, and the answer is 8.