Trirectangular Tetrahedron — Definition, Formula & Examples
A trirectangular tetrahedron is a tetrahedron (four-faced solid) where three edges meet at a single vertex and are all mutually perpendicular to each other, like the corner of a box.
A trirectangular tetrahedron is a tetrahedron possessing a vertex at which the three face angles are all right angles. Equivalently, the three edges emanating from this vertex are pairwise orthogonal, and the three faces sharing this vertex are right triangles.
Key Formula
Where:
- = Volume of the trirectangular tetrahedron
- = Length of the first perpendicular edge
- = Length of the second perpendicular edge
- = Length of the third perpendicular edge
How It Works
Imagine slicing the corner off a rectangular box with a single flat cut. The piece you remove is a trirectangular tetrahedron. The three edges along the box's edges become the mutually perpendicular edges, and the slicing plane creates the fourth face, called the hypotenuse face. Its volume is computed just like a box corner: one-sixth the product of the three perpendicular edge lengths.
Worked Example
Problem: A trirectangular tetrahedron has three mutually perpendicular edges of lengths 6, 8, and 10. Find its volume.
Apply the formula: Multiply the three perpendicular edge lengths and divide by 6.
Compute: First find the product: 6 × 8 × 10 = 480, then divide by 6.
Answer: The volume is 80 cubic units.
Why It Matters
Trirectangular tetrahedra appear naturally when you decompose cubes and rectangular prisms into simpler pieces, making them useful in 3D coordinate geometry and calculus (computing volumes via triple integrals). They also satisfy a three-dimensional version of the Pythagorean theorem known as de Gua's theorem, which relates the areas of the three right-triangle faces to the area of the hypotenuse face.
Common Mistakes
Mistake: Using the standard tetrahedron volume formula (1/3 × base area × height) and forgetting to account for the 1/2 already embedded in the triangular base, effectively computing 1/3 × a × b × c instead of 1/6 × a × b × c.
Correction: Remember that the base of a trirectangular tetrahedron is itself a right triangle with area (1/2)ab. Multiplying by (1/3)c gives (1/6)abc, not (1/3)abc.