How to Calculate a 30-60-90 Triangle

Two 30-60-90 triangles placed back-to-back form an equilateral triangle.

A triangle with the angles of 30, 60 and 90 degrees is by definition a right triangle because one of the angles is 90 degrees--a right angle. Such triangles figure heavily in trigonometry instruction, so it is worthwhile to know both what the lengths of the sides of such a triangle are as well as how they can be derived.

Instructions

1. Orient the 30-60-90 triangle so that the middle-length side is horizontal on the bottom and the shortest side is on the right. Then the 30-degree angle is on the left and the 60-degree angle is on top. Denote the length of the hypotenuse with the letter H.

2. Determine the length of the shortest side by dividing H by 2. Determine the length of the bottom side by multiplying H by √3/2. Alternatively, determine the length of the bottom side by multiplying the shortest side by √3, which may be easier to remember than the number √3/2.

3. Determine H if one of the other sides is given by multiplying the shortest side by 2 or by multiplying the middle-length sides by 2/√3. Of course, if you know two sides, you're allowed to use the Pythagorean Theorem to find the third because it's a right triangle.

4. Derive where the previous numbers come from as follows. Place two 30-60-90 triangles of the same size back to back, with their middle-length sides touching in the middle and their shortest sides forming a straight line on the bottom. Note that these two triangles now form a triangle with all the angles equal to 60 degrees. The triangle is therefore equilateral. Because the angles are all the same, the lengths are all the same. So the three sides are of length H. Note especially that the bottom side is of length H. Because the bottom is composed of the two shortest-sides, the shortest side of a 30-60-90 triangle is H/2. By the Pythagorean Theorem, the middle side must be of length H√3/2.

Tips & Warnings

• The sides of a 30-60-90 triangle with length 1 hypotenuse frequently figure into trigonometric exercises. If you place the triangle inside a unit circle such that the shortest side sits on the positive x-axis and the hypotenuse of length 1 stretches from the origin to the unit circle, the point of intersection on the unit circle has x-coordinate 1/2 and y-coordinate √3/2. These are called the cosine and sine of 30 degrees. If the triangle is turned so the middle-length side sits on the positive x-axis instead, then the point of intersection on the unit circle has x-coordinate √3/2 and y-coordinate 1/2. One then says that the cosine of 60 degrees is 1/2 and the sine of 60 degrees is √3/2.

• By similar reasoning, the sine and cosine of 45 degrees are both √2/2 = 1/√2 because a 45-45-90 degree triangle with unit hypotenuse has sides of length 1/√2. Notice that therefore as you go from 30 to 45 to 60 degrees, the cosine decreases from √3/2 to √2/2 to √1/2 (=1/2), and the sine increases from √1/2 to √2/2 to √3/2. This pattern provides a helpful mnemonic for the numbers discussed in Steps 1 to 3.

• Don't confuse the above-discussed triangle with the 3-4-5 right triangle, which has a simple ratio of the sides to each other but doesn't have the same angles as the 30-60-90 right triangle.

References

• University of Georgia: Two Special Right Triangles
• Clark University: Sines and Cosines for Special Right Triangles
• University of California, Davis: Special Right Triangles