# 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

Radius of Circle 2 = r

Diameter of Circle 1 = d

Diameter of Circle 2 = d 2

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

Formula to find area of circle = π r

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

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