Square Inside a Semicircle - University of Georgia
[Pages:2]Square Inside a Semicircle
Problem: Given a semicircle with an inscribed square of side . Let be the length of the diameter on each side of the square. We want to find the ratio of to . What is ? Solution: The above problem could be translated into the diagram here
Now, let's join a vertex of the square and the center of the circle. The segment is the radius of the semicircle.
As a result, we have a right triangle. Here the radius of the semicircle, ...........(using Pythagorean formula)
Hence,
. However,
as well. Setting them equal to each other, we get
So,
(the Golden Ratio!!!).
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