Question

5. 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.​

Answer 1

8 sides and it's called an octagon.

Explanation:

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, you know that the measure of each exterior angle must be "x-90".

Using these expressions, you can find an equation for the sum of the exterior angles:

(number of sides) * (x-90) = 360

Simplifying this equation, you get:

number of sides = 360 / (x-90)

Now you just need to find a value of x that will make the number of sides an integer.

Since you know that the sum of the interior angles of any polygon is always (n-2)*180, where n is the number of sides, you can set up another equation:

(n-2)*180 = n*x

Simplifying this equation, you get:

n = 360 / (180-x)

You want both equations to give you an integer value of n, so you need to find a value of x that makes both denominators the same.

180-x = x-90

Solving for x, you get:

x = 135

Plugging this value of x into either equation gives you:

n = 360 / (180-135) = 360/45 = 8

So, the polygon has 8 sides and is called an octagon.