30-60-90 Triangle
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)
|
The mean length of a line segment picked at random in a 30-60-90 triangle was computed by E. W. Weisstein (Aug. 5, 2010) as a complicated analytic expression involving sums of 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/Inline21.gif) |
(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.
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30-60-90 triangle