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

Question

asked Jan 25, 2021 193k views

1 Answer

Syam Kumar S · Answer 1 · 2021-01-31T09:17:59+0000

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

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).