A square sheet of paper is converted into a cylinder by rolling it along its length. What is the ratio of the base radius to the side of the square ?

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\frac{1}{2 π}

\frac{\sqrt{2}}{π}

\frac{1}{\sqrt{2 π}}

\frac{1}{π}

answer is A.

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Detailed Solution

Given, a square sheet of paper is converted into a cylinder by rolling it along its length.
Let length of each side of the square

= a

.
Base radius of cylinder

= r

.
Surface area of a cylinder

= 2 πrh

.
But height

h = a

, because the square sheet is rolled along its length.
Thus, surface area of cylinder

= 2 πra

.
Surface area of a square of side ‘a’

= a^{2}

.
Therefore,

a^{2} = 2 πra

\Rightarrow a = 2 πr

\Rightarrow \frac{r}{a} = \frac{1}{2 π}

Hence, the ratio of the base radius to the side of the square is

\frac{1}{2 π}

.
Hence, option 1 is correct.

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