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The ratio of the areas of the in-circle and circumcircle of a square is

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1:2

2:4

3:4

2:1

answer is A.

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Detailed Solution

We have to find the ratio of the area of the in-circle,

A 1

and the area of circumcircle of a square,

A 2

.
The area of a circle can be expressed as

A = π r 2

. Here

r

is the radius of a circle.
Suppose the side of a square be

x

. Then, the radius of in-circle is half the length of side of a square. Thus, the radius of in-circle,

r 1

is,

r 1 = x 2

.
Since, the radius of circumcircle is half the length of the diagonal of square.
The diagonal, d of square is,

d = x 2 + x 2 ⇒ d = 2 x 2 ⇒ d = 2 x

Therefore, the radius of circumcircle,

r 2

is,

r 2 = x 2

.
Thus, the ratio of the areas,

A 1 a n d Α 2

will be,

A 1 A 2 = π r 1 2 π r 2 2 ⇒ A 1 A 2 = π x 2 2 π x 2 2 ⇒ A 1 A 2 = 24 ⇒ A 1 A 2 = 12

Therefore, the ratio of the areas of the in-circle and circumcircle of a square is 1:2.
Therefore, the correct option is 1.

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