Question

Abcd is a square of side 2a. Fond the ratios between the circumference and the area of the incircle and the circum-circle of the square

Answer 1

the ratio of circumference is..1:√2
the ratio of area is 1:2

Answer 2

Answer: Ratio of circumference is  1:√2 and ratio of area is 1:2

Step-by-step explanation:

ABCD is a square of side 2a

Now diameter of the in-circle of square = side of square = 2a

radius of incircle = a

Diameter of the circum-circle = diagonal of square =  $\\\\\sqrt{(2a)^2+(2a)^2}= 2\sqrt{2} a$

radius of circum-circle = √2a

Now

$\dfrac{\text{circumference of incircle}}{\text{circumference of circum-circle}} = \dfrac{2\pi r_1}{2\pi r_2} = \dfrac{a}{\sqrt{2}a } =\dfrac{1}{\sqrt{2} }$

$\dfrac{\text{Area of incircle}}{\text{Area of circum-circle}} = \dfrac{2\pi r_1^2}{2\pi r_2^2} = (\dfrac{a}{\sqrt{2}a }) ^2=\dfrac{1}{2 }$

Hence, ratio of circumference is  1:√2 and ratio of area is 1:2