Question

If the ratio of circumference of two circles is 4 : 9, the ratio of their area is (1) 9 : 4 (2) 16 : 81 (3) 4 : 9 (4) 2 : 3

Answer 1

Let radii of circles be r1 and r2
given,
$2\pi \: r1:2\pi \: r2 = 4:9 \\ \: r1:\: r2 = 4:9 \\ ratio \: of \: area = \pi {r1}^{2} :\pi {r2}^{2} \\ = {r1}^{2} :{r2}^{2} = {4}^{2} : {9}^{2} = 16:81 \\ option \: \: (2)$

Answer 2

Given:

The ratio of circumference of two circles = 4 : 9

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

We have to find, the ratio of area of two circles is equal to:

Solution:

We know that,

The circumference of the circle = $2\pi r$

∴ $2\pi r_{1} :2\pi r_{2}$ = 4 : 9

⇒ $r_{1} :r_{2}$ = 4 : 9

⇒ $\dfrac{r_{1} }{r_{2} }}$ = $\dfrac{4}{9}$            ....... (1)

∴ The ratio of area of two circles = $\pi r_{1}^2 :\pi r_{2}^2$

= $r_{1}^2 :r_{2}^2$

=$(\dfrac{r_{1} }{r_{2} }})^2$

=$(\dfrac{4}{9}})^2$

= 16 : 81

∴ The ratio of area of two circles = 16 : 81

Thus, the required "option 2) 16 : 81 is correct".