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Ans is 10. Each interior angle of pentagon =108, factors=6.6.3 In order to create the complete circular all angle must be equal to 360. Now see the factors of 360=6.6.10 Now lets do the highest common factor=6.6, that means we need to split the each angle of pentagon into 3 parts and use those sides to form some other pentagon then only circular ring can be formed. that means 10 pentagon are required to create circular ring.

Let me know the source of the question, its really difficult and quite difficult to solve in the exam. +1 to you mate.

The circular ring encloses a n-sided regular polygon. Since the sum of the angles of a k-sided polygon is (n-2)180º, each angle of a regular pentagon has a degree measure of 540/5 = 108º. Thus the degree measure of each angle of the n-sided regular polygon is 360 - 2(108) = 144º. But we know that the degree measure of each angle of the n-sided regular polygon is also 180(n-2)/n, so n=10.

It is also true that a 144º angle causes a change in direction of 36º. 10 such changes are needed to yield the entire 360º needed

Well I am not sure if I could follow your solution. I used simpler method.. 108=6.6.3, while 360=6.6.10, so if we create circular ring using pentagons then interior angle of pentagon must be able to divide 360. Thats possible only when we divide the pentagon into 3 parts, ie creating angle =36. In that case we need 10 such pentagons.

I think everyone is using the same formula but doing certain steps in their head and now writing the steps down on paper and that is what is confusing. Start with the basic formula.

If we want to find the value of an interior angle of any polygon the formula is:

\(\frac{180(n-2)}{n}\)

When n = 5 for a pentagon, you get

\(\frac{180(5-2)}{5} = 108\)

So in the picture below, A & B each = 108. If you think of A + B + C as being inside a complete circle (imaginary circle) you know the total must = 360. So 360 - 108 - 108 = 144.

So Angle C in the picture is 144 degrees. This is an interior angle of a polygon formed by all the pentagons being joined. We now have to answer the question: A polygon with how many sides has interior angles of 144? Now we know the answer, but we don't have n. Before we had n =5 (pentagon) but we didn't have the answer. This is basic alegbra.