Question

The diameter of circle A is one-quarter, the diameter of circle B. The area of circle B is how many times greater than the area of circle A?

Answer 1

Hint: We will use the fact that if we take the ratio of the diameters of two circles then the ratio of the areas will be the square of the linear ratio obtained. So, as we have to find how many times the area of circle B is greater than the area of circle A and we are given the ratio of diameter so, we will apply the above concept and find the desired result.

Complete step by step answer:

We will first consider the given data that the diameter of circle A is one-quarter of the diameter of circle B.
Now, we will consider the figure,

We need to find the ratio of the areas of both circles.
As we know that all circles are similar figures, so we can use the relationship that if we take the ratio of corresponding linear dimensions like diameter then the ratio of the areas will be the square of the linear ratio.
We will let \[x\] as the diameter of circle \[A\] and accordingly the ratio gets changes.
As the ratio of the diameter of the circle, A to that of circle B is given as \[1:4\] which is converted into \[x:4x\].
So, we can find the ratio of the area of the circle which can be calculated by squaring the ratio of diameter given.
Thus, we get,
\[ \Rightarrow {\left( x \right)^2}:{\left( {4x} \right)^2} = 1:16\]
Hence, we can conclude that circle B is 16 times larger than the circle A.

Note: We can determine the area of circle A as we are given the ratio of diameters from which we can find the value of radius for both the circles as and hence find the value of the area of circle B and then we will get that the circle B is 16 times larger than the circle A. this is the alternative method to find the required answer.