The areas of two circles are in the ratio 1 ∶ 2. If the two circles are bent in the form of squares, what is the ratio of their areas?

Formula used:

Area of circle = πr²

Area of square = (side)²

Calculation:

Let the radius of smaller and larger circles be r and R respectively and the sides of smaller and larger squares be a and A respectively.

Area of smaller circle = πr² = 1

⇒ r = 1/√π 

Circumference of smaller circle = Circumference of smaller square 

⇒ 2πr = 4a

⇒ a = √π/2       

Area of smaller square

⇒ a² = π/4           ----------(1)

Area of larger circle = πR² = 2

⇒ R = √2/√π 

Circumference of larger circle = Circumference of larger square 

⇒ 2πR = 4A

⇒ A = \(\frac{\sqrt{π\ \times\ 2}}{2}\)

Area of larger square 

⇒ A² = 2π/4 = π/2    ----------(2)

from equation (1) & (2)

⇒ \(\frac{Area\ of\ smaller\ square}{Area\ of\ larger\ square}\ =\ \frac{\frac{π}{4}}{\frac{π}{2}}\ =\ \frac{1}{2}\)

∴ The correct answer is 1 : 2.