Question

If the area of a square is the same as the area of a circle, then the ratio of their perimeters of the square and that of the circle.

Answer 1

Hint: Area is the enclosed space inside a two-dimensional shape. The shape can be a polygon, such as a triangle, square, or rectangle. It can also be a curvilinear shape, like a circle. Area is always measured in square units.
Square $A = {s^2}$ s is the length of any side. ……………………………(1)
Circle $A = \pi {r^2}$ r is the radius. ……………………………(2)
Perimeter is the distance around a two-dimensional shape. For polygons, perimeter can be found using only addition by adding the distances as you move around the shape.
Square $P = 4s$ s is the length of any side. ……………………………(3)
Circle $P = 2\pi r$ r is the radius ……………………………(4)

Complete step-by-step answer:
We have been given that area of a circle of radius r is equal to the area of a square of side s.
So, with the help of equation (1) and (2)
\[
   \Rightarrow \pi {r^2} = {s^2} \\
   \Rightarrow s = \sqrt \pi r \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left( 5 \right) \\
 \]
We have to find the ratio of the perimeter of the square and that of a circle.
So, with the help of equation (3) and (4)
$
   \Rightarrow Perimeter\;of\;a\;square:Perimeter\;of\;a\;circle \\
   = 4s:2\pi r \\
 $
With the help of equation (5), we are replacing the value of s in the above equation.
$
   = 4\sqrt \pi r:2\pi r \\
   = 2\sqrt \pi :\pi \\
   = 2:\sqrt \pi \\
 $

Note: A ratio is how many times bigger one thing is than another. It's a number you multiply by to get one thing from another. But remember, when you find the ratio of two quantities, they must be in the same units.