a square is circumscribing a circle . the square has a diagonal of 30 cm.. and it has 2 quarter circle at the end of the sides of the square.. what is the area of the unused space?

Question

anonymous · Answer

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anonymous · Answer

The side of the square is $$30/\sqrt{2} = 15*\sqrt{2}$$ because its a square, so the sides of the square make up a 45-45-90 triangle.
The radius of the big circle is then that divided by two, or$$7.5*\sqrt2$$
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If we zoom in on the little circle, we can see how to solve for the radius of the small circle. Let the radius of the small circle be r, the radius of the big circle be R. Half the diagonal of the square would be $$R+r+r*\sqrt2$$. That last little distance is r*sqrt(2) because you can connect the radius to make a square, and its just another 45-45-90 triangle. R we know from earlier, 7.5*sqrt(2)
Then you solve $$7.5*\sqrt2+r+r\sqrt2 = 15$$ to get little r. The unused space is then just the area of the square minus the areas of the circles.
I got $$r=15\sqrt2 - 20$$