Cayley–Menger Triangle Geometry
Recover triangle area from distances and detect impossible distance data.
Free theory articleRecover triangle area from distances and detect impossible distance data.
Concept & interpretation
The Cayley-Menger determinant provides a way to calculate the volume (or area, in two dimensions) of a simplex using only the pairwise distances between its vertices. For a triangle, this allows computing its area directly from the lengths of its three sides, effectively generalizing Heron's formula and offering a robust method to check if given distances can actually form a valid geometry.
Experiment
Open the Cayley–Menger Triangle Geometry module to test how different distances affect the determinant and the resulting geometry.
- Select the Sides control and set it to Impossible.
- Observe the computed Area squared value. It becomes negative, which means the area would be an imaginary number.
- Notice the Status changes to Impossible and the visual display explicitly warns of impossible margins rather than trying to draw a broken shape.
- Change the Scale control to 2 and observe how the area scales quadratically (e.g., from 6 to 24 for the Right triangle preset).