Triangle Dissection

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 n=2, 3, ... smaller triangles (Vicher).

A triangle partition is prime if it does not contain a triangle partition of lower order. The number of prime triangle partitions of orders n=2, 3, ... are 1, 1, 3, 8, 62, 535, 4213, ... (OEIS A053740).


A specific type of triangle dissection consists of triangle DeltaABC together with an interior point P such that the original side lengths and the additional three segments created by connecting the triangulation point with the vertices are all integers. An example of such a dissection is illustrated above (Pegg).


Other possible dissections allow cut lines to be drawn from arbitrary points along the sides. Allowing only primitive triangles without any parallel lines, isosceles triangles, or similar triangles, the smallest 3-piece integer dissection of each of the four possibly types is illustrated above.


Similarly, the smallest 4-piece dissection for each of the 23 possible 4-piece dissections is shown above.


Two 5-piece dissections are illustrated above (Pegg).

See also

Integer Triangle, Square Dissection, Triangle Dissection Paradox

Explore with Wolfram|Alpha


Hoggatt, V. E. Jr. and Denman, R. "Acute Isosceles Dissection of an Obtuse Triangle." Amer. Math. Monthly 68, 912-913, 1961.Pegg, E. Jr. "Triangles.", E. Jr., E. Jr. "Dividing an Integer Triangle Into Smaller Integer Triangles.", N. J. A. Sequences A053740 and A056814 in "The On-Line Encyclopedia of Integer Sequences."Vicher, M. "Triangle Partitions."

Referenced on Wolfram|Alpha

Triangle Dissection

Cite this as:

Weisstein, Eric W. "Triangle Dissection." From MathWorld--A Wolfram Web Resource.

Subject classifications