By sharing adjacent sides, exactly n regular pentagons can form a circ

Thanks . The Answer is 10

I didnt completely follow your method on factorization.

I got this in my friends GMAT notes. But couldnt understand the last step

Let me mention the technque

Interior angle of a pentagon = 108 deg by using formula ( n-2)x180 / n ; where n is no of sides

Now I didnt follow hence forth

360 - ( 108 x2 ) = 144 ( Why )

Hence interior angle of pentagon in middle is 144.

Now using same formula n-2 (180) /n = 144

we get n = 10.

Hence there are 10 pentagons needed

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

the line between centers of neighbor pentagons change its direction by 1/2*360/5=36. Therefore, we need 360/36=10 pentagons to complete a circle.

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.

\(\frac{180(N-2)}{n}=144\)
180(N-2) = 144N
180N - 360 = 144N
36N - 360 = 0
36N = 360
N = 10

So we have created a 10-sided polygon by joining all of those pentagons together.

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