Question

If the perimeter of a circle and a square are equal, then what is the ratio of the area of the circle to that of the square?
1. 1 ∶ π
2. 2 ∶ π
3. 3 ∶ π
4. 4 ∶ π

Answer 1

Correct Answer - Option 4 : 4 ∶ π

Given: 

The perimeter of the circle is equal to that of the square.

Formula:

If “r” is the radius of the circle and a” is the side of the square,

1) Perimeter of a circle = 2πr

2) Area of a circle = πr²

3) Perimeter of a square = 4a

4) Area of a square = a²

Calculation:

Let “r” be the radius of the circle and “a” be the side of the square,

According to the question,

The perimeter of the circle is equal to that of the square.

2πr = 4a

⇒ r/a = 4/(2π) = 2/π      ----(1)

Now, the ratio of the area of circle and Area of square:

⇒ πr²/a²      ----(2)

From equation (1) and (2),

πr²/a² = π(2/π)²

⇒ πr²/a² = 4/π

∴ The ratio of the area of a circle to that of a square is 4 ∶ π.