Question

The ratio of the areas of incircle and circumcircle of a square is

Answer 1

let the side of square be 2x
 So the radius of the incircle will be x
   and the area of the incircle=3.14 *x*x

and the diameter of the circumcircle=2x√2
 and the radius = x√2
and area will be = 3.14*x√2*x√2

so the ratio of area=$\frac{3.14 *x*x}{3.14*x√2*x√2} =1:2$

Answer 2

Answer:

Ratio of areas of incircle and circumcircle of a square = 1:2

Step-by-step explanation:

To find,

The ratio of the areas of incircle and the circumcircle of a square.

Recall the concepts.

Incircle of a square.

The sides of the square are the tangents to the circle.

The diameter of the incircle = length of the side of  the square.

Circumcircle of a square

The circle passes through all the four sides of the square.

The diameter of the circumcircle = length of the side of  the square.

Let 's' be the side of the square

Then, we have diameter of the incircle = s

Radius of the incircle = $\frac{s}{2}$

Area of the incircle = π( $\frac{s}{2}$)² = $\frac{1}{4}$πs²

The diameter of the circumcircle = diagonal of the square = √2s

Radius of the circumcircle = $\frac{s\sqrt{2} }{2}$ = $\frac{s}{\sqrt{2} }$

Area of the circumcircle =  π($\frac{s}{\sqrt{2} }$)² =  $\frac{1}{2}$πs²

Ratio of areas of incircle and circumcircle =  $\frac{1}{4}$πs² :  $\frac{1}{2}$πs²

=   $\frac{1}{4}$ :  $\frac{1}{2}$

= 1:2

∴Ratio of areas of incircle and circumcircle of a square= 1:2

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