Open Metric
Science, Fiction and Me

Today I Learned

Dimension Analysis and Pythagorean Theorem

We can derive Pythagorean theorem using simple dimension analysis.

Triangles

The first step is to construct the formula for areas. By dimension, we know that area should be length squared. On the other hand, we also know that the area of a triangle is determined given the hypotenuse and an angle. Here we use the example of triangle ABC, whose area should be written as

\[\begin{equation} S_{\text{ABC}} = C \times \mathrm{AC}^2 \times f(\angle \mathrm{BAC}), \end{equation}\]

where $f(\cdot)$ is a dimensionless function and $C$ is a contant.

Then we write down the area for triangle ABC, triangle ABD, triangle BCD.

\[\begin{align} S_{\text{ABC}} &= C \times \mathrm{AC}^2 \times f(\angle \mathrm{BAC}), \\ S_{\text{ABD}} &= C \times \mathrm{AB}^2 \times f(\angle \mathrm{BAC}), \\ S_{\text{BCD}} &= C \times \mathrm{BC}^2 \times f(\angle \mathrm{CBD}) \\ &=C \times \mathrm{BC}^2 \times f(\angle \mathrm{BAC}). \end{align}\]

We also know that

\[\begin{equation} S_{\text{ABC}} = S_{\text{ABD}} + S_{\text{BCD}}, \end{equation}\]

which becomes

\[\begin{equation} \mathrm{AC}^2 = \mathrm{AB}^2 + \mathrm{BC}^2. \end{equation}\]

A Random #TIL# for You


Most technical TILs will be updated at datumorphism.leima.is/til

By OctoMiao

Last updated