Question

If the circumferences of two circles are in the ratio 4 : 9, then the ratio in their area is

Answer 1

Gɪᴠᴇɴ :-

If the circumferences of two circles are in the ratio 4 : 9.

ᴛᴏ ғɪɴᴅ :-

Ratio of the areas of two circles

sᴏʟᴜᴛɪᴏɴ :-

We know that,

➡ Circumference of Circle = 2πr

Let radius of 1st circle be (r) and 2nd circle be (R)

Then,

• Circumference of 1st Circle = 2πr

• Circumference of 2nd Circle = 2πR

➡ Ratio = 2πr/2πR = r/R = 4/9. ---(given)

➭ r/R = 4/9. --(1)

Now,

➡ Area of Circle = πr²

• Area of 1st Circle = πr²

• Area of 2nd Circle = πR²

➡ Ratio = πr²/πR² = r²/R² = (r/R)² --(2)

Then,

Put the value of (1) in (2) , we get,

➭ (r/R)² = (4/9)²

➭ r²/R² = 16/81

Hence,

• Ratio of areas of circles = 16:81

Answer 2

Radius of first circle = R1

Radius of second circle = R2

$\frac{area \: of \:first \: circle }{area \: of \: second \: circle} = \frac{4}{9} \\ \frac{\pi \:R {1}^{2} }{\pi \:R2² } = \frac{4}{9} \\ \frac{R1²}{R2²} = \frac{2²}{3²} \\ \frac{R1}{R2} = \frac{2}{3}$

$\frac{circumference \: of \: first \: circle }{circumference \: of \: second \: circle } = \frac{2\pi \: R1}{2\pi \: R2} \\ \\ \frac{circumference \: of \: first \: circle }{circumference \: of \: second \: circle } = \frac{r1}{r2} \\ \\ \frac{circumference \: of \: first \: circle }{circumference \: of \: second \: circle } = \frac{2}{3}$