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The area of a circle inscribed in a square of side $a$ , is $a^{2} {cm}^{2}$ .With justification, state whether the statement is:

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True; since the radius is

\frac{a}{2} cm

False; since the radius is

\frac{a}{2} cm

True; since the radius is

\frac{a}{4} cm

False; since the radius is

\frac{a}{4} cm

answer is B.

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Detailed Solution

Given: Area of the circle

= a^{2} {cm}^{2}

And, the side of the square

= a cm

Therefore, the diameter of the circle will also be equal to the side of the square as the circle is inscribed in the square.

\Rightarrow

Diameter

= a cm

\Rightarrow 2 \times radius = a cm

\Rightarrow radius = \frac{a}{2} cm

Therefore, according to the radius we have found, now we will be finding the area of the circle then we will be matching it with the given area whether it is true or false.
So, for the radius of the circle

= \frac{a}{2} cm

Area of the circle

= π \times {(radius)}^{2}

= π \times {(\frac{a}{2})}^{2}

= π \times \frac{a^{2}}{4}

= \frac{π a^{2}}{4} {cm}^{2}

Now, as we can see that the area of the circle given in the question is

a^{2} {cm}^{2}

, but we have found the area of the circle as

= \frac{π a^{2}}{4} {cm}^{2}

, which are not the same.
Hence, for the radius of the circle be

\frac{a}{2} cm

, the area of the circle cannot be

a^{2} {cm}^{2}

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