30-60-90 triangle

A 30-60-90 triangle is a right triangle having interior angles measuring 30°, 60°, and 90°. The figure below shows a 30-60-90 triangle example.

What is a 30 60 90 triangle

A 30-60-90 triangle is a type of right triangle. It is considered a special right triangle because it has predictable and consistent sides and angle measures, which enables us to use shortcuts to determine all the sides and angles of the triangle given enough information. This is useful in both geometry and trigonometry.

In the case of a 30-60-90 triangle, only 1 side needs to be known in order to find all the lengths of the other sides.

How to know if a triangle is 30 60 90

To know whether a triangle is a 30-60-90 triangle, we only need to know two of the angles. Since the sum of the internal angles of a triangle is 180°, if we know that the angles of a triangle are 30 60, 30 90, or 60 90, then the triangle is guaranteed to be a 30-60-90 triangle.

30 60 90 triangle properties

The following are some properties of a 30-60-90 triangle.

• The shortest side of a 30 60 90 triangle is opposite the 30° angle; the longer side is opposite the 60° angle; the hypotenuse, the longest side, is opposite the 90° angle.


• The 3 sides of a 30 60 90 triangle always exist in the same ratio.


• All 30 60 90 triangles are similar; in other words, 30 60 90 triangles only vary in size. They share the same ratios and angles, as depicted in the figure below.

  

How to solve a 30 60 90 triangle

To solve a 30-60-90 triangle we use the 30 60 90 triangle rule, also referred to as the 30 60 90 triangle theorem.

30 60 90 triangle rule

Given a single side of a 30-60-90 triangle, it is possible to find all the other sides of the triangle since the ratio always remains the same. 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.

Thus, given a 30 60 90 triangle and one side of the triangle, we can find the other two sides as follows:

• Given the length of the hypotenuse, divide the hypotenuse by 2 to find the shorter leg. Then, multiply the shorter leg by to find the longer leg.


• Given the length of the longer leg, divide by to find the shorter leg. Then multiply the shorter leg by 2 to find the hypotenuse.


• Given the shorter leg, multiply by 2 to find the hypotenuse and multiply by to find the longer side.

Below are some examples of how to find the sides of a 30-60-90 triangle.

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:

30 60 90 triangle theorem proof

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:

Area of a 30 60 90 triangle

The area of a 30-60-90 triangle can be found using a variation of the standard formula . Since we know the ratios of the length of the sides of a 30-60-90 triangle, we can find the area of a 30-60-90 triangle using the formula:

Perimeter of a 30 60 90 triangle

The perimeter of a 30-60-90 triangle can be simplified since the ratio of the sides is known. The perimeter of a 30-60-90 triangle can be found using the perimeter formula:

where a is the length of the short side of the triangle.