Question

the area of two circles are in the ratio 25 : 36 .find the ratio of their circumference

Answer 1

ANSWER WITH STEPS AND FULL EXPLANATION

Let the radii of the two circles be $r_{1}$ and $r_{2}$ respectively. 

Finding the areas of the two circles
Area of the circle with radius $r_{1}$ = $\pi r_{1}^{2}$
Area of the circle with radius $r_{2}$ = $\pi r_{2}^{2}$

Finding the circumferences of the two circles
Circumference of the circle with radius $r_{1}$ = $2 \pi r_{1}$
Circumference of the circle with radius $r_{2}$ = $2 \pi r_{2}$

Now, it is given in the question that the ratio between the areas of the two circles is $25:36$. 
∴ $\pi r_{1}^{2} : \pi r_{2}^{2} = 25:36$
⇒ $\frac{ \pi r_{1}^{2}}{ \pi r_{2}^{2}} = \frac{25}{36}$
⇒ $\frac{r_{1}^{2}}{r_{2}^{2}} = \frac{5^{2}}{6^{2}}$
⇒ $(\frac{r_{1}}{r_{2}})^{2} = (\frac{5}{6})^{2}$
⇒ $\frac{r_{1}}{r_{2}} = \frac{5}{6}$                           ...(1)

Now, the ratio between the circumferences of the two circles should be $2 \pi r_{1} : 2 \pi r_{2}$. 

Now, 
$2 \pi r_{1} : 2 \pi r_{2}$
$=\frac{2 \pi r_{1}}{2 \pi r_{2}}$
$=\frac{r_{1}}{r_{2}}$
$=\frac{5}{6}$                                               [ Using (1) ]
$=5:6$

∴ The ratio between the circumferences of the two circles is $5:6$. 

Hope this may help you. 

If you have any doubt, then you can ask me in the comments. 

Answer 2

Answer:

A1:A2=25:36

Divide by π:

R12:R22=25:36

Taking square root of both sides:

R1:R2=5:6

Multiplying by 2π:

2πR1:2πR2=5:6

C1:C2=5:6