Question

A circle touches the sides of the square ABCD. BEFG is a square of

side 1. The length of AB is​

Answer 1

As shown in the figure, the circle is inscribed in square

A

B

C

D

, and square

E

F

G

H

is inscribed in the circle.

Let

O

be the center of the circle, and

r

1

be the radius of the circle.

⇒

O

,

F

and

N

form a right triangle.

By Pythagorean theorem,

(

r

1

)

2

=

(

r

2

)

2

+

(

r

2

)

2

=

2

(

r

2

)

2

side length of square

A

B

C

D

=

2

r

1

side length of square

E

F

G

H

=

2

r

2

⇒

Area

A

B

C

D

: Area

E

F

G

H

=

(

2

r

1

)

2

:

(

2

r

2

)

2

=

4

(

r

1

)

2

:

4

(

r

2

)

2

=

8

(

r

2

)

2

:

4

(

r

2

)

2

=

2

:

1