Question

A square sheet of paper is converted into a cylinder by rolling it along its side. What is the ratio of the base radius to the side of the square?
a)$1:5\pi $
b)$3:2\pi $
c)$1:\pi $
d)$1:2\pi $

Answer 1

Hint: When a sheet of paper is rolled, the pair of opposite sides will join. Therefore, when the paper is rolled and the opposite sides are joined, then the base will be in the form of a circle and its circumference would be equal to the side of the square. Taking the side of the square sheet to be a variable x, we can find the radius of the base in terms of that variable and then by dividing it with the side length, we can find out the required ratio.

Complete step-by-step answer:
In this question as the value of the side length or the base radius is not given, we can take the base radius of the rolled paper to be r.
As the base of the rolled paper will be in the form of a circle and the circumference of a circle of radius r is given by the formula
$\text{Circumference of the circle of radius r}=2\pi r................(1.1)$
Then, when the paper is rolled and joined along the sides, the circumference of the base would be equal to the side length of the square.
Thus, from equation (1.1), the side length of the square sheet should be equal to $2\pi r$.
Thus the ratio of the base radius (r) to the side length of the square ($2\pi r$) should be
$\dfrac{\text{Base Radius}}{\text{Side length of square}}=\dfrac{r}{2\pi r}=\dfrac{1}{2\pi }$
Which matches the answer in option(d). Thus the correct answer is option(d) which has the value $\dfrac{1}{2\pi }$.

Note: In this case we could also have taken the side length of the square as x and then found out the base radius as $\dfrac{x}{2\pi }$ using equation (1.1). However, the ratio would be the same as obtained earlier.