Remix: 30 60 90 Triangle


30 60 90 Triangle

30 60 90 triangle

Properties of a 30-60-90 triangle

A 30-60-90 triangle is characterized by its three angles (30°, 60°, 90°) and side lengths (x:2x:√3x). Since the side lengths are related to each other in a specific way, we can infer that this triangle is “special.”

Knowing this information, we can calculate the area of a 30-60-90 triangle by using the formula for the area of any triangle: Area = 1/2 base x height. The base is x, and the height can be calculated using trigonometry: h = x * sin(60°). Therefore, the area of any 30-60-90 triangle will always be equal to 3/2x².

Characteristics of a 30-60-90 triangle

There are three Characteristics of a 30-60-90 triangle:

Sides of a 30-60-90 triangle

At the heart of a 30-60-90 triangle is its sides. Each side has a distinct length and angle measures in at the following ratios:

• Opposite angle to 90 degrees = 1/2 hypotenuse

• Opposite angle to 60 degrees = hypotenuse

• Opposite angle to 30 degrees = 2 x hypotenuse

These sides give the 30-60-90 triangle unique attributes, which come together perfectly. This is why it stands out from other triangles, such as equilateral triangles. 

To draw a 30-60-90 triangle, you only need to draw two consecutive angles of 30 and 60 degrees connected by a line. This will be the hypotenuse, the longest side of the triangle. The other two sides (opposite to angles 30 and 60 degrees) are in a ratio of 1:2, respectively.

Angles of a 30-60-90 triangle

The angles of a 30-60-90 triangle measure precisely 30 degrees, 60 degrees, and 90 degrees. 

These three angles also form the basis for many mathematical properties and formulas related to this triangle. For instance, the Pythagorean Theorem is a popular formula that ties into this triangle; it states that the sum of two sides squared equals the hypotenuse squared.

Ratios of the sides of a 30-60-90 triangle

The sides of a 30-60-90 triangle also have a distinctive ratio. The length of the hypotenuse is double that of the side opposite to the 60-degree angle and half of the side opposite to the 90-degree angle. This makes it easy to calculate all three sides when given one, which is incredibly useful in many real-life applications.

In addition to that, the ratio of any two sides is always a perfect square number (1, 4, 9…). This makes it easier to calculate and work with the triangle’s angles.

30 60 90 Triangle FAQs

What is the rule for 30, 60 90 triangles?

The 30-60-90 triangle is a special right triangle, meaning that one of its angles is 90 degrees. The other two angles measure precisely 30 and 60 degrees, which are in the ratio of 1:2:3. This also relates to the sides length of this triangle; the side opposite the 30-degree angle will be half as long as the hypotenuse and the side opposite the mid-sized degree angle will be twice as long as the hypotenuse.

How do you solve a 30-60-90 triangle with one leg?

If you have one side length of the 30-60-90 triangle, it is fairly simple to solve for the other two acute angles. First, calculate the hypotenuse by multiplying the known side by √3. Then, take half of that number as your answer for the side opposite the 30-degree angle.

What is special about a 30-60-90 triangle?

The 30-60-90 triangle is special because it has a fixed ratio between the angles and sides. This ratio simplifies calculations, as knowing one side length can determine the exact side lengths.

What is the formula for the triangle ratio?

The formula for the triangle ratio is often written as a/b=c/2, where a and b are the two known sides of the triangle, and c is the hypotenuse. This can also be expressed using trigonometric ratios: sin(30)=a/c and sin(60)=b/c.

You can also calculate a 30-60-90 Triangle with a 30 60 90 Triangle Calculator.


I hope you now understand how interesting and useful the 30-60-90 triangle is! From finding side lengths to solving real-life problems, this unique shape has a wide range of applications - all thanks to its ratio between angles and sides. Please be sure to familiarize yourself with the formulas for calculating each side to take advantage of this special type of triangle.