In the figure, a circle is inscribed in a square of side 5 cm and another circle is circumscribing the square. Find the ratio of the area of the outer circle to that of the inner circle.

Given,

Side of square = 5 cm.

Let A₁ and A₂ be the areas of the inner and outer circle respectively.

Inner circle (inscribed in the square):

Diameter of the circle = side of the square

Diameter = 5 cm

So, radius = $\dfrac{5}{2}$ = 2.5 cm.

Area of inner circle (A₁) = πr²

= π(2.5)²

= 6.25π cm²

Outer circle (circumscribing the square)

Diameter of the outer circle = Diagonal of the square

Diameter of the outer circle = $a\sqrt{2} = 5\sqrt{2}$ cm.

Radius of the outer circle = $\dfrac{5\sqrt{2}}{2}$

Area of outer circle (A₂) = πr²

= $π\Big(\dfrac{5\sqrt{2}}{2}\Big)^2$

= π × $\dfrac{25 × 2}{4}$

= 12.5π cm².

Ratio of areas :

Outer area : Inner area

A₂ : A₁

12.5π : 6.25π

2 : 1.

Hence, ratio = 2 : 1.