30-60-90 Triangle Calculator

It is the right triangle you get by cutting an equilateral triangle in half. An equilateral triangle has three 60-degree angles and three equal sides; drop the altitude from any vertex perpendicular to the opposite side and it splits the triangle into two congruent right triangles with angles 30, 60, and 90. That construction is why the 30-60-90 is called the half-equilateral triangle, and it is the cleanest way to see where the side ratio comes from: the long leg of the half is the altitude of the original equilateral, the short leg is half the original side, and the hypotenuse is one full original side.

Start from the equilateral construction above. Let the original equilateral side be 2; then the half-side (the short leg of the new right triangle) is 1, and the hypotenuse (still a full original side) is 2. Find the long leg with the Pythagorean theorem: long^2 + 1^2 = 2^2, so long^2 = 3, so long = sqrt(3). That is the 1 : sqrt(3) : 2 ratio, with the short leg opposite the 30-degree angle (the smallest angle gets the smallest side), the long leg opposite the 60-degree angle, and the hypotenuse opposite the 90-degree angle. Scaling the short leg to any value a scales every side by the same factor, so all 30-60-90 triangles are similar to this one.

Because both its angles and its side ratio are exact and easy to remember. Together with the 45-45-90 (the half-square, with side ratio 1 : 1 : sqrt(2)) it is the pair of right triangles that comes up so often in geometry, trigonometry, and exam questions that the ratios are worth memorizing. Both give exact values for sin, cos, and tan at the angles 30, 45, and 60 degrees without needing a calculator: sin(30) = 1/2, cos(30) = sqrt(3)/2, tan(30) = 1/sqrt(3), and the matching values for 60 follow from sin(60) = cos(30) and cos(60) = sin(30). Most other right triangles need decimal approximations; these two do not.

Draw an equilateral triangle and drop the altitude from one vertex to the opposite side. The altitude bisects that side and creates two 30-60-90 triangles. In each half, the short leg is half the original side, the long leg is the original side times sqrt(3) / 2, and the unchanged original side is the hypotenuse. This is also the geometry of a 30-60-90 drafting set square.

In any setting that uses 30-degree or 60-degree angles. Drafting and architecture use the 30-60-90 set square for isometric pictorials (the three axes are 30 degrees above horizontal, 30 degrees below horizontal, and vertical), for hex-shaped plans, and for stairs whose stringers are set at a 30-degree or 60-degree pitch. Hex grids in tile work, board games, and 3D printing are built from rows of 30-60-90 triangles; every hexagon decomposes into six equilateral triangles and therefore twelve 30-60-90 right triangles. Civil engineering uses it for cut-and-fill slopes specified as 1:sqrt(3), and surveyors use it to set out 60-degree property corners without a transit. The 1 : sqrt(3) : 2 ratio lets you size every side from any single known measurement with one multiplication or division.