Question

Eight circles of radius 1 have centers on a larger common circle, and adjacent circles are tangent. Find the area of the common circle.

Answer 1

THE AREA OF COMMON CIRCLE IS 2 ( 2 + √ 2 ).

Step-by-step explanation:

As given in the information let us consider eight small circles each having radius 1 and are adjacent to each other

These adjacent circles are tangent to each other.

These 8 small circles form together a big circle which is having radius 'R' to small circles.

Let us consider the angle between big circle and small circle is $\alpha$.

$\alpha$ = $\pi$ for one circle

As we are having eight circles angle for eight circles is

$\alpha$ = $\pi$ /8

Radius R = 1 / sin $\alpha$

cos 2$\alpha$ = 1 - 2sin²$\alpha$ => sin 2$\alpha$ = (1- cos2$\alpha$)/2 and with cos  $\pi$ /8 = 1/2 * √ 2

we get 1/sin  $\pi$ /8  = 4/(2- √ 2)

Therefore area of common circle is

=> 2(2+√ 2).