Question

a) 24 cm
If perimeter of a square is equal to the circunference of a circle, then its ratio of areas is
a) : 2
b) 1:4
c) 1:3
d) 1:1​

Answer 1

➲$\underline{\underline{\pink{\sf Given:}}}$

• Perimeter of a square is equal to the circumference of Circle

➲$\underline{\underline{\pink{\sf Find:}}}$

• Ratio of Areas of Circle and Square

➲$\underline{\underline{\pink{\sf Given:}}}$

Let, the radius of circle = r

$\sf\therefore \underline{ so, we \: know \: that}$

$\sf \to circumfrence = 2 \pi r$

________________

$\sf \leadsto perimeter \: of \: square = 4a$

Now, it is given that the circumference of Circle is same as Perimeter of Square.

So,

$\sf \to perimeter \: of \: square = 4a$

$\sf : \implies 2 \pi r= 4a$

$\sf : \implies a = \dfrac{2 \pi r}{4}$

$\sf : \implies a = \dfrac{\pi r}{2}$

Hence, side of square will be $\sf \dfrac{\pi r}{2}$

____________

Now,

$\hookrightarrow\sf Ratio = \dfrac{area \: of \: circle}{area \: of \: square}$

$\hookrightarrow\sf Ratio = \dfrac{ \pi {r}^{2} }{{(side)}^{2} }$

where,

• Radius = r
• π = 22/7
• side = $\sf \dfrac{\pi r}{2}$

So,

$\hookrightarrow\sf Ratio = \dfrac{ \pi {r}^{2} }{{(side)}^{2}}$

$\hookrightarrow\sf Ratio = \dfrac{ \pi {r}^{2} }{{(\dfrac{ \pi r}{2} )}^{2}}$

$\hookrightarrow\sf Ratio = \dfrac{ \pi {r}^{2} }{{\dfrac{ {\pi}^{2}{r}^{2}}{4}}}$

$\hookrightarrow\sf Ratio = \pi {r}^{2} \times \dfrac{4}{ {\pi}^{2} {r}^{2} }$

$\hookrightarrow\sf Ratio = 1 \times \dfrac{4}{\pi}$

$\hookrightarrow\sf Ratio = \dfrac{4}{\pi}$

$\hookrightarrow\sf Ratio = \dfrac{4}{ \dfrac{22}{7} }$

$\hookrightarrow\sf Ratio =4 \times \dfrac{7}{22}$

$\hookrightarrow\sf Ratio =2 \times \dfrac{7}{11}$

$\hookrightarrow\sf Ratio = \dfrac{14}{11}$

$\hookrightarrow\sf Ratio = 14 : 11$

_________________

Hence, the Ratio of Area of square and Circle will be 14:11