A 30-60-90 triangle is a right triangle having angles of 
, 
, and 
. For a 30-60-90 triangle with hypotenuse
of length 
,
the legs have lengths
 |  |  |
(1)
|
 |  |  |
(2)
|
and the area is
 |
(3)
|
|
|
The inradius 
and circumradius 
are
 |  |  |
(4)
|
 |  |  |
(5)
|
Pegg et al. (to appear) enumerated the dissections of a 30-60-90 triangle into smaller similar triangles.
The numbers of arrangements with 
, 2, ... pieces are 1, 1, 3, 18, 121, 844, 6397, 53145, ...
(OEIS A400462), where the first term counts
the uncut triangle.
The mean line segment length between two points chosen independently from the uniform distribution on a 30-60-90 triangle was computed by E. W. Weisstein (pers. comm., Aug. 5, 2010) as a complicated analytic expression involving sums of natural logarithms. After simplification, the result can be written as
 |  | ![1/(1440)[204+36sqrt(3)+81ln3+2(9+8sqrt(3))ln(2+sqrt(3))]a](/images/equations/30-60-90Triangle/Inline22.svg) |
(6)
|
 |  |  |
(7)
|
(E. Weisstein, M. Trott, A. Strzebonski, pers. comm., Aug. 25, 2010; OEIS A180308).
30-60-90 triangles are used in drafting, as illustrated above. This allows lines of 
,
,
,
and 
to be drawn by sliding the drafting triangle along a
T-square.
Pegg, E. Jr. Mathematical Games. Episode 46: "Sporadic Similar
Triangles." Oct. 22, 2026.