How to Calculate the Diagonal Length of a Cube

The Organic Chemistry Tutor

This concise mathematics tutorial demonstrates how to calculate the length of the space diagonal of a cube. Using a step-by-step derivation, the narrator applies the Pythagorean theorem in three dimensions to create a general formula. The video begins by drawing a cube and defining the problem, then breaks the 3D problem into two 2D triangles to solve for the unknown length. The core of the lesson focuses on deriving the formula $d = x\sqrt{3}$, where $x$ is the side length. The narrator first calculates the face diagonal of the cube's base using two side lengths. This result is then used as the base of a secondary right triangle inside the cube, allowing for the calculation of the final space diagonal. The method highlights the utility of algebraic substitution in geometry. This video is highly valuable for high school geometry classes as it connects algebraic manipulation with spatial reasoning. It moves beyond rote memorization of formulas by showing the logical proof behind them. Teachers can use this to introduce 3D coordinate geometry, reinforce the Pythagorean theorem, or as a practical example of working with radicals and square roots.