30 60 90 Triangle Ratio, Formula & Sides

What is 30 60 90 Triangle?

The 30 60 90 triangle is a special right triangle widely used in trigonometry because it has well-defined relationships between its sides and angles, which can be used to calculate trigonometric functions such as sine, cosine, and tangent with angles 30°, 60°, and 90°.

It’s also known as a bisected equilateral triangle because if you draw an equilateral triangle and draw an elevation on one of its sides, you create two 30 60 90 triangles. The sides of the triangle have a certain relationship that can be summarized as follows.

The 30 60 90 triangle side ratios are a special ratio. The side opposite the 30° angle is half the length of the hypotenuse. So the shortest side is always half of the longest side. In other words, the hypotenuse is twice the length of the side opposite the 30° angle.

The side opposite the 60° angle is (√3)/2 times the length of the hypotenuse. Or √3 times the side opposite the 30° angle.

For the side lengths of a 30 60 90 triangle formula is:

Shortest side = x
Longer side = x√3
Hypotenuse = 2x

Here is a few 30 60 90 triangle examples with answers

If the shortest side of a 30-60-90 triangle measures 4 inches, then the longer side is:

x√3 = 4√3

And the hypotenuse is:

2x = 8

So the longer side of the triangle measures 4√3 inches, and the hypotenuse measures 8 inches.

For another example, if you know that the side opposite the 30° angle is 6, you can find the lengths of the other sides as follows:

The hypotenuse is twice the length of the side opposite the 30 degree angle, so 2 x 6 = 12.
Since the side opposite the 60 degree angle is (√3)/2 times the length of the hypotenuse, (√3)/2 x 12 = 6√3.
So the sides of this 30-60-90 triangle are 6, 6√3, and 12.

So if you know the side length of a 30 60 90 triangle, you can use this ratio to find the lengths of the other two sides. Conversely, if we know the length of any two sides, we can use these relations to find the length of the third side. This relationship is derived from the Pythagorean Theorem, which states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse.

30-60-90 triangles are useful in geometry and trigonometry because they have predictable ratios between their side lengths, which can be used to solve for unknown values. It has real applications in fields such as engineering, architecture and construction. For example, it can be used to design and build ramps, stairs and other structures with gradual slopes. Its properties and applications make it a valuable tool in solving geometric problems and in the design of real-world structures.