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If the diameter of a circle is equal to the diagonal of a square, then the ratio of their areas is,

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7 : 1

1 : 1

11 : 7

22 : 7

answer is C.

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Detailed Solution

It is given that the diameter of a circle is equal to the diagonal of a square. We need to find the ratio of their areas.
Let the diameter of the circle be

d

units and side of the square be

a

units.
We know that the length of the diagonal of a square with side

a

units is

\sqrt{2} a

units [because of Pythagoras theorem].
By data,

d = \sqrt{2} a

— (1)
We know that the area of a circle is

π r^{2}

sq.units, where

r

is the radius of the circle.
We know that,

d = 2 r

\Rightarrow Area of circle = π (\frac{d}{2})^{2}

\Rightarrow Area of circle = π \frac{d^{2}}{4}

— (2)
We know that the area of a square with side

a

units is

a^{2}

sq.units.

\Rightarrow Area of square = a^{2}

Substituting the value of

a

from eq(1),

\Rightarrow Area of square = {(\frac{d}{\sqrt{2}})}^{2}

\Rightarrow Area of square = \frac{d^{2}}{2}

— (3)
From eq(2) and eq(3),

\frac{Area of circle}{Area of square} = \frac{(π \frac{d^{2}}{4})}{(\frac{d^{2}}{2})}

\Rightarrow \frac{Area of circle}{Area of square} = π \frac{d^{2}}{4} \times \frac{2}{d^{2}}

\Rightarrow \frac{Area of circle}{Area of square} = \frac{π}{2}

\Rightarrow \frac{Area of circle}{Area of square} = \frac{22}{7} \times \frac{1}{2}

[because

π = \frac{22}{7}

]

\Rightarrow \frac{Area of circle}{Area of square} = \frac{11}{7}

Therefore, if the diameter of a circle is equal to the diagonal of a square, then the ratio of their areas is

11 : 7

.
Hence, the correct option is

3

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