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30-60-90 Triangle

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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)

Pegg et al. (2026) enumerated the similar-triangle dissections of a 30-60-90 triangle. The numbers of arrangements with n=1, 2, ... pieces are 1, 1, 3, 18, 121, 844, 6397, 53145, 460404, ... (OEIS A399999), where the first term counts the uncut triangle. This sequence was a runner-up in the OEIS 400K celebration contest (OEIS Foundation Inc. 2026).

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

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 Dissection, Triangle Line Picking

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References

OEIS Foundation Inc. "A400K Winners." 2026. https://oeis.org/wiki/A400K_Winners. Pegg, E. Jr. "Winners of the OEIS 400K Milestone Contest." Wolfram Community. Oct. 26, 2026. https://community.wolfram.com/t/28423. Pegg, E. Jr. Mathematical Games. Episode 46: "Sporadic Similar Triangles." Oct. 22, 2026. https://www.youtube.com/watch?v=bp7mOio0ef0.Pegg, E.; Trease, B.; Hudson, A.; and McNeil, D. S. "How Does One Cut a 30-60-90 Triangle?" 31 Oct 2026. To appear in Geombinatorics. https://arxiv.org/abs/2611.03719.Sloane, N. J. A. Sequences A180308 and A399999 in "The On-Line Encyclopedia of Integer Sequences."

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30-60-90 Triangle

Cite this as:

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

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