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Geometry Puzzle: Find the Area of the Square from Circle Diameter

A circle and a square. The top vertices os the square lie in the circle. The bottom side of the square is tangent to the circle. A horizontal double arrow represents the diameter of the circle and is labeled/labelled ten.

A circle has diameter 10.

What is the area of the square?

Solution

As the circle has diameter 10, its radius is 5.

Let xx be the square’s side length.

Consider the right(-angled) triangle below.

Its hypotenuse is 5, the radius of the circle. The horizontal side is half of the square’s side length and its vertical side measures 𝑥−5\textit{x}-5.

A circle and a square. The top vertices os the square lie in the circle. The bottom side of the square is tangent to the circle. A right/right-angled triangle has vertices the center/centre of the circle, the top vertex of the square and the middle point of the top side of the square. Its horizontal side, vertical side and hypotenuse are labeled/labelled x over two, x minus five and five, respectively. A vertical double arrow represents the distance from the top side of the square and the circle and is labeled/labelled ten minus x.

Using the Pythagorean/Pythagoras’ theorem:

52=(x−5)2+(x2)25^2=\left(x-5\right)^2+\left(\frac{x}{2}\right)^2
25=x2−10x+25+x2425=x^2-10x+25+\frac{x^2}{4}
0=x2−10x+x240=x^2-10x+\frac{x^2}{4}
0=4x2−40x+x20=4x^2-40x+x^2
5x2−40x=05x^2-40x=0
5x(x−8)=05x\left(x-8\right)=0
x=0,x=8x=0, \, x=8

Then square’s length is

x=8x=8

So, the area of the square is

828^2
6464

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