Question

in the image given below the radius of all the large circles is R then what is the radius of the inner smaller circle​

Answer 1

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According to the question, the radius of all the large circles is r. Let us assume rn as the radius of the inner smaller circle and connect the radius of all of the circles, as shown below. Thus, we can say that the radius of the inner smaller circle is (√2 - 1)r.

Answer 2

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If we look at the figure carefully, we notice that

AB = BC = 2r

AC = 2r + 2rn

Now, in right angle triangle ΔABC

AC2 = AB2 + BC2

⇒ AC2 = (2r)2 + (2r)2

⇒ AC2 = 4(r)2 + 4(r)2

⇒ AC2 = 8(r)2

⇒ AC = r√8

⇒ AC = r√(2 × 2 × 2

⇒ AC = 2r√2

⇒ 2r + 2rn = 2r√2

⇒ 2rn = 2r√ 2 - 2r

⇒ 2rn = 2r(√2 - 1)

⇒ rn =  2r(√2 -1)2  

⇒ rn = (√2 - 1)r

Thus, we can say that the radius of the inner smaller circle is (√2 - 1)r.

Learn More:-

If the circles share a common center point, then the distance between the inner circle and the outer circle is the difference of their radii.

But if the center points of the two circles are different, it gets a little more complicated.

Let AA be the center point of the smaller circle with radius r,r, and let BB be the center point of the larger circle with radius R.