Square Root Form

Let's take an example where a=10 (base); what would the hypotenuse be?

Multiply 10 with sqrt(2) and we get 14.14…

\(10*\sqrt{2} = 14.14\)

Take the area of the smaller sided square 100 (10×10) and the area of the greater perimeter square (14.14)² should equal twice the area of the larger line segment of the two.

And it is! (199.94, 0.06 difference to due rounding error)

Here's an image to feast your mind! I will be inclined to add 3D mathematical proofs when I finish the research.

Prompt. Gemini v.1.0.7* 6 Aug. 2025 [at 4PM PST], https://g.co/gemini/share/2a0fc51fe2e3