Question 65

If the circumference of a circle is equal to the perimeter of a square, and the radius of the given circle is positive, then which of the following options is correct?

If the circumference of a circle is equal to the perimeter of a square.

circumference of a circle = perimeter of a square

$$2\times\ \pi\ \times\ radius\ =\ 4\times\ length\ of\ each\ side$$

$$2\times\ \frac{22}{7}\times\ radius\ =\ 4\times\ length\ of\ each\ side$$

$$\frac{11}{7}\times\ radius\ =\ \ length\ of\ each\ side$$
$$\frac{11}{7}=\frac{length\ of\ each\ side}{radius}$$
Let's assume the radius and length of each side are 7y and 11y respectively.

Here the radius of the given circle is positive, then we need to find out the relation between the area of circle and square.

area of circle = $$\pi\ \times\left(radius\right)^2$$

= $$\frac{22}{7}\times\left(7y\right)^2$$

= $$\frac{22}{7}\times49y^2$$

= $$22\times7y^2$$

= $$154y^2$$    Eq.(i)

area of square = $$(length\ of\ each\ side)^2$$

= $$(11y)^2$$

= $$121y^2$$    Eq.(ii)

Eq.(i) > Eq.(ii)

$$154y^2$$ > $$121y^2$$

So Area of the circle > Area of the square

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