Question

How can you convert the area of ${A_R} = l \times b$ of a rectangle into the area of a circle?

Answer 1

Area is the region covered by the object in the two-dimensional plane. Area of the rectangle can be expressed as the product of length and the breadth whereas the area of the circle is the product of the pi and radius squared. Here we will equate the area of the rectangle and the area of the circle and will find the unknown term in terms of the radius “r”.

Complete step-by-step solution:
Let us assume that the area of the rectangle is ${A_R} = l \times b$ …. (A)
Similarly, the area of circle be ${A_C} = \pi {r^2}$ …. (B)
Where, $l$is the length of the rectangle
is the breadth of the rectangle
And is the radius of the circle
From the given word statement, the area of the rectangle is equal to the area of the circle.
Therefore, using equations (A) and (B)
${A_R} = {A_C}$
Place values in the above equation,
$l \times b = \pi {r^2}$
The above equation can be re-written as
$\pi {r^2} = l \times b$
Make radius the subject and move other terms on the opposite side. When the term multiplicative on one side is moved to the opposite side, then it goes to the denominator.
$\Rightarrow {r^2} = \dfrac{{l \times b}}{\pi }$
Take the square root on both the sides of the equation.
$\Rightarrow \sqrt {{r^2}} = \sqrt {\dfrac{{l \times b}}{\pi }}$
Square and square root cancel each other on the left-hand side of the equation.
$\Rightarrow r = \sqrt {\dfrac{{l \times b}}{\pi }}$
This is the required solution.

Note: Be careful while doing the simplification, when you move any term from one side to another. When any term is multiplicative on one side then it goes to the denominator. But if it is in the addition with another term on one side then it becomes negative when moved to the opposite side.