The circumference of a circle is 10pi. Which of the following is not

There is no limit on how small this rectangle can be, but the biggest rectangle that can fit inside would be the square that is inscribed in the circle. Therefore we can just pick the biggest choice here as there is only 1 option that is incorrect.

Ans: E

If you need the math: The radius of the circle is 5, this corresponds to half of the diagonal of the square inscribed in the circle. Then the square's area is 10*5 = 50. Choice E is the biggest so it is most suspicious, \(40*\sqrt{2} > 40*1.4 > 50\) so E is out. The next biggest choice is D, which is about 30*1.4 = 42 so it is within 50.
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I took a different approach
from the given circumference we can find out the radius and thus the diagonals of rectangle
diagonals divide the rectangle into 4 isosceles triangles
each side will be equal to 5(root2) and thus max area will be 50

since option E is beyond 50, it cannot be the area of rectangle inscribed in a circle

Radius of the circle=5. Therefore Area is 25pi = 78.5
E) 40 root 2 > 78.5

Think about it - what is the constraint on a rectangle inside a fixed circle?

It can have very very little area (say the blue rectangle). Then we can start expanding it till it becomes the yellow rectangle (which is actually a square). If we try to expand it any more, its area will start decreasing again till it becomes very very small again (the green rectangle).




Hence, there is no minimum area of the rectangle but there is a maximum possible area dependent on the value of the radius. So if 40 is an acceptable area of a rectangle, then 30 is certainly acceptable too. If 30*sqrt(2) is acceptable (which is about 42), then 40, 30 and 20*sqrt(2) are certainly acceptable too. Hence, the only value that would not be acceptable is the greatest value given i.e. 40*sqrt(2).

Answer (E)
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