If the measure of an interior angle of a regular polygon is 165, find the sum of the measures of its interior angles
sum of angles of a regular polygon = 180(n-2)
so 180(n-2)/n = 165
180n - 360 = 165n
15n = 360
n = 24
sum of inside angles = 180(22) = 3960°
To find the sum of the measures of the interior angles of a regular polygon, we can use the formula:
Sum of Interior Angles = (n - 2) * 180
where 'n' is the number of sides of the polygon.
In this case, we are given that the measure of an interior angle is 165. Since the sum of the measures of the interior angles of a polygon is determined by the number of sides, we need to find the number of sides.
To do this, we can use the formula for the measure of an interior angle of a regular polygon:
Interior Angle = (n - 2) * 180 / n
We plug in the given angle measure (165) into this formula and solve for 'n'.
165 = (n - 2) * 180 / n
To eliminate the fraction, we can cross-multiply:
165n = (n - 2) * 180
Expanding the right side:
165n = 180n - 360
Subtracting 180n from both sides:
-15n = -360
Dividing both sides by -15:
n = 24
Therefore, the polygon has 24 sides.
Now we can use the sum formula to find the sum of the measures of its interior angles:
Sum of Interior Angles = (n - 2) * 180
= (24 - 2) * 180
= 22 * 180
= 3960
Hence, the sum of the measures of the interior angles is 3960 degrees.