Try this beautiful problem from Geometry: Radius of a circle
A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?
Geometry
Cube
square
Answer: $ \frac{2}{\sqrt \pi} $
AMC-8 (2005) Problem 25
Pre College Mathematics
The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square
Can you now finish the problem ..........
Region within the circle and square be \(x\) i.e In other words, it is the area inside the circle and the square
The area of the circle -x=Area of the square - x
can you finish the problem........
Given that The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square
Let the region within the circle and square be \(x\) i.e In other words, it is the area inside the circle and the square .
Let r be the radius of the circle
Therefore, The area of the circle -x=Area of the square - x
so, \(\pi r^2 - x=4-x\)
\(\Rightarrow \pi r^2=4\)
\(\Rightarrow r^2 = \frac{4}{\pi}\)
\(\Rightarrow r=\frac{2}{\sqrt \pi}\)
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