A Diagonal Inequality

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.



This inequality can be proved by realizing that the square roots in the inequality are the lengths of the diagonals of the four cuboids shown.

Specifically, three cuboids , and of dimensions , and fit inside the cuboid of dimensions corner to corner. Call the corners , , and . The diagonal of cuboid is , and the length of the broken line is the sum of the diagonals of , and , which is , due to the extended triangle inequality.


Contributed by: Ed Pegg Jr (August 2022)
Open content licensed under CC BY-NC-SA




[1] N. Stojanović, "Proof without Words: Inequality for the Sum of the Diagonals," The Mathematical Intelligencer, 43(4), 2021 p. 32. doi:10.1007/s00283-021-10075-9.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.