Question

A circular wire of radius 42 cm is bent to form a square. find the ratio of their areas

Answer 1

Given,
Radius of the circle is r
Therefore, area of the circle =pi*r^2
Also,perimeter of the circle =2pi*r
Perimeter of the circle = perimeter of the square
Side of the square = 2pi*r/4
=pi*r/2
Area of the square =(pi*r/2)^2
=pi^2*r^2/4
Ratio=area of circle /area of square
=(pi*r^2)/(pi^2*r^2/4)
=4/pi

Hence, the ratio is 4/pi...
Sorry, if the answer seems a bit too clumsy.

Answer 2

Radius of circle = 42cm

$circumference \: of \: circle = 2\pi \: r \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 2 \times \frac{22}{7} \times 42cm \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =264cm$

Perimeter of Square = 264 cm
[ as the same wire is bent into sq. ]

$side \: of \: sq. = \frac{perimeter}{4} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{264}{4} cm \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 66cm$

$area \: of \: sq. = side \times side \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (66 \times 66)cm^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 4356 {cm}^{2}$

$area \: of \: circle \: = \pi \: {r}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{22}{7} \times 42 \times 42 {cm}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 5544 {cm}^{2}$

$ratio \: of \: their \: areas \: = \frac{4356}{5544} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{11}{14} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

♦ Ratio of their areas = 11:14