The circle has a radius 1. The diagonal of the square extends to be a tangent of the circle. The circle touches the black lines at three points as shown. Find the area of the square.
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Solution
By constructing an inner square in the right-hand side of the circle and calling the length of the square x, we get
Using the principle of equal tangents, the tangent from the diagonal of the square equals the tangent extending from the base of the square.
The tangent in blue has length 1 + x.
Zooming in on the diagram, we can use the same principle for the small pair of tangents shown in green as below.
The length of the green tangent is the side length of the square subtracted by the radius of the circle, x -1.
Therefore the diagonal of the square is the blue tangent minus the green tangent.
We know that the side of the square is x.
Using the Pythagorean Theorem
which is the area of the square!
And that’s our answer. How amazing. Hey, don’t forget to clap👏 the article as a token of appreciation. Thank you 🦁
