a circle and a square have the same area if we denote the circumference of the c...

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August 2021 1 9 Report

a circle and a square have the same area if we denote the circumference of the circle and P is the perimeter of the square, then

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SOLUTION

TO CHOOSE THE CORRECT OPTION

A circle and a square have the same area if we denote C the circumference of the circle and P is the perimeter of the square, then

1. P > C

2. P = C

3. P < C

4. P = 2C

EVALUATION

Let r be the radius of the circle and a be the side of the square

Since circle and a square have the same area

$\therefore \sf{ \: \: \pi {r}^{2} = {a}^{2} } \: \: \: ......(1)$

C the circumference of the circle

$\sf{C = 2\pi r}$

P = 4a

$\therefore \sf{ \: \: {C}^{2} = 4 {\pi}^{2} {r}^{2} }$

$\sf{ { P}^{2} = 16 {a}^{2} }$

$\implies \: \sf{ { P}^{2} = 16 \pi \: {r}^{2} }$

$\therefore \: \: \displaystyle \sf{ \frac{ {P}^{2} }{ {C}^{2} } = \frac{4}{ \pi} }$

$\implies \: \displaystyle \sf{ \frac{ {P}^{2} }{ {C}^{2} } = \frac{28}{ 22} } > 1$

$\implies \sf{P > C}$

FINAL ANSWER

Hence the correct option is

1. P > C

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