A triangle dissection divides a triangle into finitely many smaller triangles with nonoverlapping interiors.
The following table summarizes variants and their distinguishing features.
| variant | scope |
| Triangle Dissection--Equilateral Triangle | an equilateral triangle divided into mutually similar pieces or pieces of equal area |
| Triangle Dissection--Exceptional Shapes | special triangle shapes with exceptional arrangement counts |
| Triangle Dissection--Integer Triangles | an integer triangle with integer triangle pieces |
| Triangle Dissection--Similar Triangles | mutually similar pieces and enumeration of arrangements |
| Triangle Dissection--Six-Piece Geometry | generic non-right triangle; six pieces similar to the original |
Hoggatt and Denman (1961) showed that any obtuse triangle can be divided into eight acute isosceles triangles.
There are 1, 4, 23, 180, 1806, 20198, ... (OEIS A056814) topologically distinct ways to divide a triangle into
, 3, ... smaller triangles
(Vicher).
Cipra (2026) enumerated 21 equilateral triangle dissections into four triangles of equal area, up to rotations and reflections.
A triangle partition is prime if it does not contain a triangle partition of lower order. The numbers of
prime triangle partitions of orders 
, 3, ... are 1, 1, 3, 8, 62, 535, 4213, ... (OEIS
A053740).
