To find the number of sides in a regular polygon based on its interior angle, we can use the formula:
Interior Angle = ((n – 2) × 180) / n
where n is the number of sides of the polygon.
1. When the interior angle is 160°:
Let’s set the interior angle to 160°:
160 = ((n – 2) × 180) / n
Multiplying both sides by n:
160n = (n – 2) × 180
Expanding the right side:
160n = 180n – 360
Rearranging gives:
360 = 180n – 160n
360 = 20n
Now, dividing by 20:
n = 18
Thus, if the interior angle is 160°, the polygon has 18 sides.
2. When the interior angle is 150°:
Now, let’s set the interior angle to 150°:
150 = ((n – 2) × 180) / n
Again, multiplying both sides by n:
150n = (n – 2) × 180
Expanding the right side:
150n = 180n – 360
Rearranging gives:
360 = 180n – 150n
360 = 30n
Now, dividing by 30:
n = 12
Thus, if the interior angle is 150°, the polygon has 12 sides.