Question

If the ratio of areas of circles is $4:9:25$, what is the ratio of their radii?

Answer 1

Hint: The radii given here means the plural of radius, since many circles are being talked about this means that the word would be radii, now we do know that the relation between the area and the radius of the circle, the area of the circle is given by,
$A = \pi {r^2}$,
Thus we see that the area is related to the square of the radius, thus whatever be the ratio of the circle with their area the radius will be the square root of that.

Complete step by step solution:
The first step would be find the relation between the area and the radius of a circle when two circles are given, then we will rewrite the formula for three circles as is given in the question, lets compare the area of two circles,
$\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{\pi {r_1}^2}}{{\pi {r_2}^2}}$
Upon simplifying by cancelling $\pi $ we get,
$\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{r_1^2}}{{r_2^2}}$
Which gives,
${r_1}:{r_2} = \sqrt {{A_1}} :\sqrt {{A_2}} $
Scaling it for three circles we get,
${r_1}:{r_2}:{r_3} = \sqrt {{A_1}} :\sqrt {{A_2}} :\sqrt {{A_3}} $
Now we will get the radii as,
\[{r_1}:{r_2}:{r_3} = \sqrt 4 :\sqrt 9 :\sqrt {25} \]

The value of the ratio of the radii of the three circles upon solving the root sign will be,
${r_1}:{r_2}:{r_3} = 2:3:5$

Note:
In case the question asks about the ratio of the radii of the circle when the circumference instead of the area is given it will be written as,
${r_1}:{r_2} = {C_1}:{C_2}$
Where ${C_1}$ and ${C_2}$ are the circumferences of the two circles.