Question

What is the diameter of a circle whose area is equal to the sum of the areas of two circles of diameter \[10cm\] and \[24cm\]?

Answer 1

Hint: Here we are asked to find the diameter of a circle with the given data. We are given the data that the area of the big circle is the sum of the areas of two small circles. First, by using the formula of area of a circle we will find the areas of those two circles then adding them and equating them with the required area of the circle we will find its diameter.
Formula Used: Formula that we need to know before solving the problem:
Area of a circle: \[\pi {r^2}\]
\[r - \]the radius of a circle

Complete step-by-step solution:
It is given that the area of a big circle is the sum of the areas of the two small circles of diameter \[10cm\] and \[24cm\]. We aim to find the diameter of the circle.
First, we will find the area of the two small circles.
To find the area of the first small circle:
We know the area of a circle: \[\pi {r^2}\] where \[r\] the radius of the circle.
Here the diameter of a circle is given \[10cm\]. Let the area of this small circle be \[{A_1}\].
Thus, \[{A_1} = \pi \times {5^2}\]
On simplifying this we get \[{A_1} = 25\pi \]
To find the area of the second small circle:
We know the area of a circle: \[\pi {r^2}\] where \[r\] the radius of the circle.
Here the diameter of a circle is given \[24cm\]. Let the area of this small circle be \[{A_2}\].
Thus, \[{A_2} = \pi \times {\left( {\dfrac{{24}}{2}} \right)^2}\]
On simplifying this we get \[{A_2} = 144\pi \].
Now we have found the area of two small circles. Also, it is given that the area of the big circle is the sum of the areas of the two small circles. Let \[A\] be the area of the big circle.
Therefore, \[A = {A_1} + {A_2}\]
We already found the values of small circles substituting them in the above we get
\[A = 25\pi + 144\pi \]
On simplifying this we get
\[A = 169\pi \]
We know that the area of the circle is written as \[\pi {r^2}\]. Thus, the above expression will become \[\pi {r^2} = 169\pi \] where \[r - \] is the radius of a required circle.
On simplifying the above, we get
\[{r^2} = 169\]
\[\Rightarrow r = \sqrt {169} \]
\[\Rightarrow r = 13\]
Thus, we have found the value of the radius of the big circle. So, the required diameter of the big circle will be \[26cm\].

Note: We know that the diameter is nothing but twice the radius of that circle. That is if \[r\] is the radius of a circle then the diameter of the circle will be \[d = 2r\]. In the above calculation, we have skipped this step to make the calculation short.