In a regular polygon each exterior angle is 90° less than each interior angle. Calculate the number of sides of the polygon hence give its name.
Let's begin by using the fact that the sum of the exterior angles of any polygon is always 360 degrees.
Let's call the measure of each interior angle "x". Then, we know that the measure of each exterior angle must be "x-90".
Using these expressions, we can find an equation for the sum of the exterior angles:
(number of sides) * (x-90) = 360
Simplifying this equation, we get:
number of sides = 360 / (x-90)
Now we just need to find a value of x that will make the number of sides an integer.
Since we know that the sum of the interior angles of any polygon is always (n-2)*180, where n is the number of sides, we can set up another equation:
(n-2)*180 = n*x
Simplifying this equation, we get:
n = 360 / (180-x)
We want both equations to give us an integer value of n, so we need to find a value of x that makes both denominators the same.
180-x = x-90
Solving for x, we get:
x = 135
Plugging this value of x into either equation gives us:
n = 360 / (180-135) = 360/45 = 8
So the polygon has 8 sides and is called an octagon.