Question

the sum of circumferences of four small circles of equal radius is equal to the circumference of a bigger circle.Find the ratio of the bigger circle of that of smaller circle​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\therefore{\text{Ratio\:of\:radius=4:1}}}$

$\green{\therefore{\text{Ratio\:of\:Area=16:1}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{ \underline \bold{Given : }} \\ : \implies \text{Small \:circle \: radius = r} \\ \\ : \implies \text{Big \: circle \: radius = R} \\ \\ : \implies \text{Circumference \: of \: four \: circle = Circumference \: of \: big \: circle} \\ \\ \red{ \underline \bold{To \: Find : }} \\ : \implies \text{Ratio \: of }\: \: \: \frac{R}{r} = ?$

• In the given question information given about the sum of circumferences of four small circles of equal radius is equal to the circumference of a bigger circle and we have to find the ratio of the bigger circle of that of smaller circle.

• According to given question :

$: \implies \text{Circumference \:of \: small \: circle = }2\pi \: r \\ \\ : \implies \text{Circumference \: of \: 4 \: small \: circle = 4 }\times \pi \: r \\ \\ \bold{Again,} \\ : \implies Circumference \: of \: big \: circle = 2\pi \: R \\ \\ : \implies 8\pi r = 2\pi \: R\\ \\ : \implies \frac{8\pi r}{2\pi } = R \\ \\ : \implies R = 4r \\ \\ \bold{For \: Ratio \: of \: their \: radii}\\ : \implies Ratio \: of \: their \: radi = \frac{Big \: circle }{Small \: circle} \\ \\ \green{: \implies Ratio = \frac{R}{r} = \frac{4r}{r} = \frac{4}{1} } \\ \\ \bold{For \: Ratio \: of \: their \: area's}\\ : \implies Ratio \: of \: their \: area = \frac{Area \: of \: big \: circle}{Area \: of \: small \: circle} \\ \\ :\implies Ratio = \frac{\pi {R}^{2} }{\pi {r}^{2} } \\ \\ : \implies Ratio = \frac{ {4r}^{2} }{ {r}^{2} } \\ \\ \green{: \implies Ratio = \frac{16r}{r} = \frac{16}{1} }$

Answer 2

Correct Question :--- the sum of circumferences of four small circles of equal radius is equal to the circumference of a bigger circle.Find the ratio of the area bigger circle of that of area of one smaller circle.. ?

Concept And Formula :---

→ Circumference of a circle with radius r is given by 2πr ..

→ Area of a circle with radius r is given by πr².

→ Ratio of Circumference of 2 circle with Radius as r1 and r2 is given by = r1:r2.

→ Ratio of Area of 2 circle with Radius as r1 and r2 is given by = (r1)² : (r2)²

[ Reason :-- As π will cancel From both sides we get, only Radius Result in Ratios. ]

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Solution :---

Let radius of Each smaller circle be = r

and, radius of big circle be = R.

According to Question Now,

→ Circumference of one Small circle = 2πr

→ Circumference of 4 Small circle = 4*2πr = 8πr .

So,

→ Circumference of Big circle = Circumference of 4 Small circle

→ Circumference of Big circle = 8πr

→ 2πR = 8πr

Dividing both sides by 2π we get,

→ R = 4r .

_________________________

Now, Either we put This value in Formula of Area for Required ratio, or we can simply , Square the ratios .

Lets Try both :---

→ Area of one smaller circles = πr²

→ Area of big circle = π(R²)

Putting value of R = 4r now, we get,

→ Area of big circle = π(4r)² = 16πr²

So,

Required ratio =

Area of big circle : Area of one small circles = 16πr² : πr²

→ 16πr² : πr²

Dividing both sides by πr² we get

→ 16 : 1 (Ans.)

____________________

Now, lets see it by above told concept .

Area Ratio = (r1)² : (r2)²

→ smaller circle radius = r

→ bigger circle radius = R

And,

→ R = 4r

so,

→ Area of big circle : Area of one small circles = (R)² : (r)²

Putting value ,

→ Area of big circle : Area of one small circles = (4r)² : r²

→ Area of big circle : Area of one small circles = 16r² : r²

→ Area of big circle : Area of one small circles = 16 : 1

__________________________

Hence, ratio of Area of bigger circle to one smaller circle will be 16:1 ..