> [!summary] The Pythagorean theorem is a principle that relates the three sides of a right triangle. > Key equation: $c^2 = a^2 + b^2$ >[!info]+ Read Time **⏱ 1 min** # Definition The Pythagorean theorem is a fundamental theorem that relates the three sides of a right triangle. It states that the area of the square of the hypotenuse is the sum of the areas of the squares of the other two sides. Mathematically, this is described by the equation and seen in the image below. $ c^2 = a^2 + b^2 $ ![[pyt_1.png|300]] [^1] > [!note] Explanation The Pythagorean theorem can be seen as the area of a square. ## Proof of the Pythagorean Theorem > [!warning] Assumption To prove the Pythagorean theorem using [[Algebraic Proof|algebraic proof]] assume the following: > - A big square with an area $(a+b)^2$ > - A big square can be filled by 4 triangles and a smaller square $ \begin{array}{c} \text{Suppose we have a sqaure with area $A_{\text{Total}} = (a+b)^2$} \\ \text{We could fill the sqaure with a smaller sqaure + 4 trangles} \\ A_{triangle} = \frac{1}{2} ab \\ 4\cdot A_{traingle} = 2ab \\ A_{\text{sqaure}} = c^2 \\ \\ \text{So our total area could be written as $A_{\text{Total}} = 4A_{\text{sqaure}} + A_{\text{sqaure}} $} \\ \text{Or:} \\ (a+b)^2 = 2ab + c^2 \\ a^2+ \cancel{2ab}+b^2 = \cancel{2ab}+c^2 \\ a^2 + b^2 = c^2 \end{array} $ ![[py_2.gif]] [^2] > [!note] Explanation Small video explanation of using logic and algebra to derive the Pythagorean theorem to accompany the derivations above # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/VjI4LtotC2o?si=xKIJBCeJ7Yy4N2lc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. --- [^1]: Photo taken from: Wikipedia contributors. (2025, May 14). _Pythagorean theorem_. https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/File:Pythagorean.svg [^2]: Photo taken from: Wikipedia contributors. (2025, May 14). _Pythagorean theorem_. https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/File:Animated_gif_version_of_SVG_of_rearrangement_proof_of_Pythagorean_theorem.gif