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 ).

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).