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

Eight circles of radius 1 have centers on a larger

victoriakraus6599

22.01.2022

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).

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habsofarah0

20.03.2023

∠x = 92°

Step-by-step explanation:

Supplementary means that the angles add up to 180°. Since one of them equals 88°, the other one or ∠x equals:

180-88 = 92

So, ∠x = 92°

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