Problem:
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let the radius of the circle be . Then the area of the circle is . The area of the square is , so its side length . The ratio of the side length of the square to the radius of the circle is .
Note: The "squaring of a circle" is a classical problem. In the latter part of the th century it was proven that a square having an area equal to that of a given circle cannot be constructed with the standard tools of straightedge and compasss because it is impossible to construct a transcendental number, e. . .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions