If the radius of a circle is r and the side of a square is x, and the circle is inscribed in the square, then find the remaining area left in the square.If the radius of a circle is r and the side of a square is x, and the circle is inscribed in the square, then find the remaining area left in the square.
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Step-by-step explanation:
The area of the first square can be
calculated if side is known.
So, we have length of diagonal
2R=
2
a (a = side length)
⇒a=
2
R
and the radius of next circle
will be
2
a
=
2
R
and So on
thus the side length of every
inscribed square will be ,
2
1
times
the previous square's side length
So, the area will be
2
1
×
2
1
times
the previous square's area so
the sum will be,
S
n
=(
2
R)
2
+(
2
2
R
)
2
+(
4
2
R
)
2
+......(
2
n−1
2
R
)
n→∞
lim
S
n
=(
2
R)
2
⎣
⎢
⎢
⎡
1−
2
1
1
⎦
⎥
⎥
⎤
=2×2R
2
=4R
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