Question

If the circumference of a circle and the perimeter of a square are equal, then which of the following statements is correct?
$(a)$ Area of circle = Area of square
$(b)$ Area of the circle > Area of the square
$(c)$ Area of circle < Area of the square
$(d)$ Nothing definite can be said about the relation between the areas of the circle and the square.

Answer 1

Hint: Use the direct formula for circumference of the circle which is $\left(2\pi r\right)$ where r is the radius of the circle and perimeter of the square which is 4a if a is the side of the square to obtain the relation between the side of square a and the radius of circle r.

Complete step-by-step answer:
As we know that the circumference of a circle is calculated as $\left(2\pi r\right)$ where r is the radius of the circle.
Now let the side of the square be a unit.
As we know that square has four sides and all have equal lengths.
Now we know that the perimeter of any shape is the sum of all side lengths.
So the perimeter of square = (a + a + a +a) = 4a
Now it is given that the circumference of the circle is equal to the perimeter of the square.
$\Rightarrow 2\pi r=4a$...................................... (1)
Now as we know that the area (AC) of the circle is $\left(\pi r^2\right)$ and the area (AS) of the square is side square (i.e. $a^2$).
$\Rightarrow A_C=\pi r^2$ and $A_S=a^2$
Now from equation (1) the value of r in terms of a is
$\Rightarrow r={\displaystyle \frac{4a}{2\pi }}={\displaystyle \frac{2a}{\pi }}$
Now substitute this value of r in area of circle we have,
$\Rightarrow A_C=\pi r^2=\pi {\left({\displaystyle \frac{2a}{\pi }}\right)}^2={\displaystyle \frac{4a^2}{\pi }}$
Now as we know that the value of $\left[\pi =3.14\right]$ so $\left[{\displaystyle \frac{4}{\pi }}>1\right]$
Therefore,
$\Rightarrow {\displaystyle \frac{4a^2}{\pi }}>a^2$
$\Rightarrow A_C>A_S$
So the area of the circle is greater than the area of the square.
Hence option (B) is correct.

Note: It is always advised to cross check each and every option while solving such types of questions. Some of the basic formulas of area of circle and area of the square need to be remembered. Perimeters in case of many figures can directly be termed as the sum of all sides. Circumference refers to the outer length of the circle. Area is usually determined in case of 2-d figures.