30-60-90 triangle
A 30-60-90 triangle is a right triangle having interior angles measuring 30°, 60°, and 90°.
Similarity
All 30-60-90 triangles are similar.
Line segments DE and FG are perpendicular to side AC of the 30-60-90 triangle, ABC. Triangles ADE and AFG are also 30-60-90 triangles so, △ABC~△ADE~△AFG. This is true for all 30-60-90 triangles.
30-60-90 triangle side lengths
The ratio of the side lengths of a 30-60-90 triangle are:
- The leg opposite the 30° angle (the shortest side) is
the length of the hypotenuse (the side opposite the 90° angle). - The leg opposite the 60° angle is
of the length of the hypotenuse. - The hypotenuse is twice the length of the shortest side.
the length of the 
of the length of the hypotenuse.
The ratios of the sides can be calculated using two congruent 30-60-90 triangles. As shown in the figure above, two congruent 30-60-90 triangles, △ACD and △BCD, share a side along their longer leg. Since ∠BCD = ∠ACD = 30°, ∠BCA = 60°. Also ∠CAD = ∠CBD = 60°, therefore △ABC is an equilateral triangle. Given that the length of one side of the equilateral triangle is s, AB = AC = BC = s and 
. Using the Pythagorean Theorem for either triangle, 
. So:
Example:
Given that the leg opposite the 30° angle for a 30-60-90 triangle has a length of 12, find the length of the other leg and the hypotenuse.
The hypotenuse is 2 × 12 = 24. The side opposite the 60° angle is 
.
30-60-90 triangle in trigonometry
In the study of trigonometry, the 30-60-90 triangle is considered a special triangle. Knowing the ratio of the sides of a 30-60-90 triangle allows us to find the exact values of the three trigonometric functions sine, cosine, and tangent for the angles 30° and 60°.
For example, sin(30°), read as the sine of 30 degrees, is the ratio of the side opposite the 30° angle of a right triangle, to its hypotenuse.
Using the side lengths 
for the 30-60-90 triangle above:
Similarly, we can find that:
