Question

The area of a square is equal to the area of a circle. What is the ratio between the side of the square and the radius of the circle?
A. \[\sqrt{\pi }:1\]
B. $1:\sqrt{\pi }$
C. $1:\pi $

Answer 1

Hint: Here, we have been given that the area of a square and circle are equal and we need to find the ratio between the side of the square and the radius of the circle. For this, we will assume the side of the square to be ‘a’ and the radius of the circle to be ‘r’. Then we will use the formula for the area of a square given as area=${{s}^{2}}$ where s is the side of the square and area=$\pi {{R}^{2}}$ where R is the radius of the circle. Then we will put s=a and R=r in these formulas to find their respective areas and then we will keep both the areas equal. Then we will get an equation in a and r using which we will find the value of $\dfrac{a}{r}$. As a result, we will get the required answer.

Complete step-by-step solution
Here, we have been given that the area of a square and the area of a circle are equal. So, let us assume that the area of the square is ${{A}_{1}}$ and that of the circle is ${{A}_{2}}$.
Thus we can say that:
${{A}_{1}}={{A}_{2}}$ ………………….(i)
Now, since we have to find the ratio of the side of the square and the radius of the circle, let us assume the side of the square to be ‘a’ units and the radius of the circle to be ‘r’ units. Thus, we have to find:
$a:r$

Now, we know that the area of a square with side ‘s’ is given as:
$Area={{s}^{2}}$
Here, s=a and area is ${{A}_{1}}$. Thus we get:
${{A}_{1}}={{a}^{2}}$ …..(ii)
Now, we also know that the area of a circle with radius ‘R’ is given as:
$Area=\pi {{R}^{2}}$
Here, R=r and area is ${{A}_{2}}$. Thus we get:
${{A}_{2}}=\pi {{r}^{2}}$ …..(iii)
Now, if we put the value of ${{A}_{1}}$ and ${{A}_{2}}$ from equations (ii) and (iii) in equation (i) we will get:
$\begin{align}
  & {{A}_{1}}={{A}_{2}} \\
 & \Rightarrow {{a}^{2}}=\pi {{r}^{2}} \\
\end{align}$
Now, we know that we can transform this equation as:
$\begin{align}
  & {{a}^{2}}=\pi {{r}^{2}} \\
 & \Rightarrow \dfrac{{{a}^{2}}}{{{r}^{2}}}=\pi \\
\end{align}$
Now, we can write it as:
$\begin{align}
  & \dfrac{{{a}^{2}}}{{{r}^{2}}}=\pi \\
 & \Rightarrow {{\left( \dfrac{a}{r} \right)}^{2}}=\pi \\
\end{align}$
Now, taking under root both sides we get:
$\begin{align}
  & {{\left( \dfrac{a}{r} \right)}^{2}}=\pi \\
 & \therefore \dfrac{a}{r}=\sqrt{\pi } \\
\end{align}$
Thus, the ratio a:r is equal to $\sqrt{\pi }:1$.
Hence, option (A) is the correct option.

Note: Some formulas of areas of different 2-D figures are given as follows:
1. Square: $Area={{s}^{2}}$ (s is the side of the square)
2. Rectangle: $Area=l\times b$ (l is the length and b is the breadth of the rectangle)
3. Triangle: $Area=\dfrac{1}{2}\times b\times h$ (b is the base of triangle and h is the height). The area of the triangle is also given by heron’s formula given as:
$Area=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$ (a, b and c are the lengths of the sides of the triangle and s is the semi perimeter given as $s=\dfrac{a+b+c}{2}$)
4. Circle: $Area=\pi {{r}^{2}}$ (r is the radius of the circle)