Question

A circular area having a radius $20\,cm$ is divided into two equal parts by a concentric circle of radius ‘r’. The value of ‘r’ will be
A. $5cm$
B. $10cm$
C. $5\sqrt 2 cm$
D. $10\sqrt 2 cm$

Answer 1

As a circular area having a radius $20\,cm$ is divided into two equal parts by a concentric circle of radius $r$.
Concentric circles are of the inner circle and must be half the area of the outer circle because they are divided into two equal parts.
Area of circle $= \pi {r^2}$
Where, $\pi$ $= \dfrac{{22}}{7}$and $r = radius\,of\,the\,respective\,circle$

Complete step by step solution:
Given,
Radius of outer circle $= 20\,cm$
Let us denote the radius of outer circle by ‘$R$’ and radius of inner circle by ‘$r$’.
Hence, $R = 20\,cm$
Area of outer circle $= \pi {R^2}$
Area of inner circle $= \pi {r^2}$, where $\pi = \dfrac{{22}}{7}$
As the circular area is divided into two equal parts by a concentric circle, the area of the inner circle must be half the area of the outer circle.
We have,
$\pi {r^2} = \dfrac{1}{2}\pi {R^2}\,\,\,\,\,\,[Put\,R = 20\,and\,\pi = \dfrac{{22}}{7}]$
$\dfrac{{22}}{7} \times {r^2} = \dfrac{1}{2} \times \dfrac{{22}}{7} \times {\left( {20} \right)^2}$
$\pi$ will be eliminated from both sides
$\Rightarrow\,{r^2} = \dfrac{1}{2} \times 20 \times 20 \\\ \Rightarrow\,{r^2} = 10 \times 20 \\\ \Rightarrow {r^2} = 200 \\\ \Rightarrow\,r = \sqrt {200} \\\ \Rightarrow\,r = 10\sqrt 2 \,cm \\\$

Hence, the radius of the inner circle $r = 10\sqrt 2 \,cm$.

Note: In the Geometry, the objects are said to be concentric, when they share the common center. Circles, spheres, regular polyhedral, regular polygons are concentric as they share the same center point. In Euclidean Geometry, two circles that are concentric should have different radii from each other.
In such a question, an important thing is to note in which conditions the circle has been divided as here it has been divided equally.