Question

The ratio of the radii of two circles is \[2:3\]. Find the ratio of their areas. If the area of the bigger circle is \[3123\]sq. cm, find the area of the smaller circle.

Answer 1

Hint:
Here, we will use the formula of the area of a circle and find the ratio between the areas of both the circles by substituting the given ratio of their radii. Then we will substitute the area of the bigger circle in the obtained ratio and solve it further to find the required area of the smaller circle.

Formula Used:
We will use the following formulas:
1) Area of a circle with radius \[r\] is \[\pi {r^2}\]
2) \[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\]

Complete step by step solution:
Let the radius of the smaller circle be \[r\] and the radius of the bigger circle be \[R\].
According to the question, the ratio of the radii of two circles is \[2:3\].
We can see that the first ratio is lesser than the second one, showing the first circle will have a smaller radius than the second one.
Thus, the ratio of radius of the smaller circle to that of the bigger one is \[2:3\].
\[r:R = 2:3\]
Writing this as fraction,
\[ \Rightarrow \dfrac{r}{R} = \dfrac{2}{3}\]………………………………..\[\left( 1 \right)\]
Now, we know that area of the smaller circle, having the radius \[r\] is \[\pi {r^2}\] and, the area of the bigger circle having the radius \[R\] is \[\pi {R^2}\].
We will find the ratio of area of both the circles.
The area of smaller circle \[ \div \] The area of the bigger circle \[ = \dfrac{{\pi {r^2}}}{{\pi {R^2}}}\]
Canceling out the like terms from the numerator and the denominator, we get
\[ \Rightarrow \] The area of smaller circle \[ \div \] The area of the bigger circle \[ = \dfrac{{{r^2}}}{{{R^2}}}\]
Now, using the formula, \[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\], we can write this as:
\[ \Rightarrow \] The area of smaller circle \[ \div \] The area of the bigger circle \[ = {\left( {\dfrac{r}{R}} \right)^2}\]
Now, substituting \[\dfrac{r}{R} = \dfrac{2}{3}\] from equation \[\left( 1 \right)\] in the above equation, we get
\[ \Rightarrow \] The area of smaller circle \[ \div \] The area of the bigger circle \[ = {\left( {\dfrac{2}{3}} \right)^2} = \dfrac{4}{9}\]
But it is also given that the area of the bigger circle is 3123 sq. cm.
Now, let us assume the area of the smaller circle be \[a\] sq. cm.
Therefore, we can write the ratio of area of the smaller circle to that of the bigger circle as: \[\dfrac{a}{{3123}} = \dfrac{4}{9}\]
Taking the denominator from the LHS to the RHS, we get,
\[ \Rightarrow a = \dfrac{4}{9} \times 3123\]
\[ \Rightarrow a = 4 \times 347 = 1388\] sq. cm

Therefore, the required area of the smaller circle is \[1388\] sq. cm
Hence, this is the required answer.

Note:
A circle is a two –dimensional shape consisting of all the points in a plane that are at a given distance from a given point i.e. the centre. The distance between the centre and point on the plane is called the radius of the circle and it is constant. In other words, radius is the distance from the centre of the circle to any point on it. The perimeter of the circle is called its circumference and it is the distance around it, whereas, the area of a circle is the region bounded by it.