TOPICS
Search

30-60-90 Triangle


30-60-90Triangle

A 30-60-90 triangle is a right triangle having angles of 30 degrees, 60 degrees, and 90 degrees. For a 30-60-90 triangle with hypotenuse of length a, the legs have lengths

b=asin(60 degrees)=1/2asqrt(3)
(1)
c=asin(30 degrees)=1/2a,
(2)

and the area is

 A=1/2bc=1/8sqrt(3)a^2.
(3)
30-60-90I
30-60-90O

The inradius r and circumradius R are

r=1/4(sqrt(3)-1)a
(4)
R=1/2a.
(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

l^_=1/(1440)[204+36sqrt(3)+81ln3+2(9+8sqrt(3))ln(2+sqrt(3))]a
(6)
=0.2885717...a
(7)

(E. Weisstein, M. Trott, A. Strzebonski, pers. comm., Aug. 25, 2010; OEIS A180308).

Drafting triangle

30-60-90 triangles are used in drafting, as illustrated above. This allows lines of 0 degrees, 30 degrees, 60 degrees, and 90 degrees to be drawn by sliding the drafting triangle along a T-square.


See also

Isosceles Triangle, Polydrafter, Right Triangle, Triangle, Triangle Line Picking

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequence A180308 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "30-60-90 Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/30-60-90Triangle.html

Subject classifications