Question

If the perimeter of a square and the circumference of a circle are equal, then the area of the square is ________ the area of the circle.

• A. greater than
• B. less than
• C. twice
• D. half

Answer 1

The correct option is B less than

Let us consider a square of side 'a' and a circle of radius ‘r'.
Perimeter of the square = 4a
Circumference of the circle = 2 $\pi$ r

Given that the perimeter of the square is equal to the circumference of the circle,
Therefore, 4a = 2 $\pi$ r
$\Rightarrow$ a = $\frac{\pi }{2}$ r = $\frac{11}{7}$ r

Area of the square
= $a^2$ = ${\left(\frac{11}{7}r\right)}^2$= $\left(\frac{121}{49}\right)r^2$

Area of the circle
= $\pi$ $r^2$= $\frac{22}{7}{r}^2$= $\frac{22\times 7}{7\times 7}{r}^2$ = $\frac{154}{49}{r}^2$

We know that 154 > 121. Divide both sides with 49 and multiply both sides with $r^2$ . We get,
$\frac{154}{49}{r}^2>\frac{121}{49}{r}^2$

$\Rightarrow$ Area of circle > Area of square

Therefore, the area of the square is less than the area of the circle even though they have the same perimeter.