The ratio between the areas of two circles is 16:9. Find the ratio between their radii, diameters and circumferences.

Given : 

Ratio between areas of two circles  =  16 : 9

To Find : 

i) Ratio between radii of circles.

ii) Ratio between diameters of circles

iii)  Ratio between circumferences of circles

Solution :

 Lets take Circle 1 and Circle 2

Radius of Circle 1 = r ₁  

Radius of Circle 2 = r ₂  

Diameter of Circle 1 = d ₁  

Diameter of Circle 2 = d ₂ 

Area of circle 1 : Area of circle 2 = 16 : 9

Formula to find area of circle = π r ²   

 $\displaystyle \begin{array}{{>{\displaystyle}l}}
π\ r^{2}_{1} \ :π\ r^{2}_{2} \ =\ 16\ :\ 9\\
\\
\frac{π\ r^{2}_{1}}{π\ r^{2}_{2}} \ =\ \frac{16}{9}\\
\\
\frac{r^{2}_{1}}{r^{2}_{2}} \ =\ \frac{16}{9}\\
\\
\left(\frac{r_{1}}{r_{2}}\right)^{2} \ =\ \frac{16}{9}\\
\\
\ \ \frac{r_{1}}{r_{2}} \ =\ \sqrt{\frac{16}{9}} \ \\
\\
\ \ \frac{r_{1}}{r_{2}} \ \ =\ \ \frac{4}{3}\\
\\
r_{1} \ :\ r_{2} \ =\ 4\ :\ 3\\
\
\end{array}$

Ratio between radii of circles = 4 : 3

Diameter = 2 r

$\displaystyle \begin{array}{{>{\displaystyle}l}}
d_{1} \ =\ 2\times r_{1} \ \ \ \ ;\ d_{2} \ =\ 2\times r_{2}\\
\\
\frac{d_{1} \ }{d_{2} \ \ } \ =\ \frac{2\times r_{1}}{\ 2\times r_{2}}\\
\\
\frac{d_{1} \ }{d_{2} \ \ } \ =\ \frac{r_{1}}{\ r_{2}} \ \\
\\
\frac{d_{1} \ }{d_{2} \ \ } \ =\ \ \frac{4}{\ 3}\\
\\
d_{1} \ :\ d_{2} \ =\ 4\ :\ 3
\end{array}$

Ratio between Diameters of two circles = 4 : 3

Circumference of circle = 2πr

$\displaystyle \begin{array}{{>{\displaystyle}l}}
\ 2πr_{1} \ \ :\ 2πr_{2}\\
\\
\ \frac{2πr_{1}}{\ 2πr_{2}} \ =\ \frac{r_{1}}{\ r_{2}}\\
\\
\ \frac{2πr_{1}}{\ 2πr_{2}} \ =\ \ \frac{4}{\ 3}\\
\\
\ 2πr_{1} :2πr_{2} \ =\ 4\ :\ 3
\end{array}$

Ratio between circumferences of circles = 4 : 3

i) Ratio between radii of circles  =  4 : 3

ii) Ratio between diameters of circles = 4 : 3

iii) Ratio between circumferences of circles = 4 : 3