# 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 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: