Home » tutorials » Unleashing the Beauty of 30-60-90 Triangles: The Perfect Triangle Revealed # Unleashing the Beauty of 30-60-90 Triangles: The Perfect Triangle Revealed

There are various types of triangles such as obtuse triangles, isosceles, acute, equilateral, scalene, and many more. But among all of them, only some types of triangles are considered special.

Special triangles mean their sides and angles are predictable and predictable and they can be used to solve various geometrical problems.

A 30-60-90 triangle is pronounced as “thirty sixty ninety”. It is a very special type of triangle indeed.

In this article, we will understand the concept of the 30-60-90 triangle, its meaning, formula, definition, sides, area, and the rules that are applied to our special triangle.

## What is the 30-60-90 Triangle?

The 30-60-90 Triangle can be defined as a triangle where the angles are always 30, 60 and 90. As one angle is always 90, it is also known as a special right triangle.

The angles of this triangle are in a ratio 3:2:1. And the sum of its two acute angles is equal to the right angle (30 + 60 = 90), so these angles are in the ratio 2 : 1 or 1 : 2.

## Sides of a 30-60-90 Triangle

As already mentioned, it is a special triangle with particular values of lengths and angles. Therefore, the sides of 30-60-90 triangles are said to be Pythagorean triples.

The basic ratio of 30-60-90 Triangle can be expressed as follows:-

 The side opposite the 30° angle x The side opposite the 60° angle x * √3 The side opposite the 90° angle 2x

## 30-60-90 Triangle Theorem

30-60-90-Triangle Theorem can be given as:-

Statement of Triangle Theorem

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shortest side, and the length of the other side is √3 times the length of the shortest side.

### The Formula of 30-60-90 Triangle

The 30-60-90-Triangle Formula can be written as 1:√3: 2 which gives the ratio of the three sides of the Triangle.
It can also be written as 1:2:3 which gives the ratio of the three angles of the 30-60-90 Triangle​​​.

## Area of 30-60-90 Triangle

As you all know we can calculate the area of a triangle by applying the given formula:-

= ½ * b * h

Now, let us understand,

How can we apply the formula on the 30-60-90 triangle:-

Perpendicular of the triangle = x/√3

Area of the triangle = (1/2) × base × perpendicular

Area = 1/2 × x × x/√3

Area= x2/(2√3) ## Some Important facts about 30-60-90 Triangle

1. The 30-60-90 triangle is also called a special right triangle.
2. The angles of the 30-60-90 triangle are in a unique ratio of 1:2:3 and sides are in the ratio 1:√3: 2.
3. The side opposite to the angle 30° is the shortest side as 30 degrees is the smallest angle
4. The side opposite to the angle of 60° is the medium side.
5. The side opposite to the angle of 90° will always be the largest side.
6. If any one side of the 30-60-90 triangle is given, we can calculate all sides. This rule is known as the 30 60-90 triangle rule.

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## In Conclusion

The 30-60-90 triangle is a special type of triangle that has unique properties that make it a valuable tool in mathematics.

Its fixed ratios of sides and angles allow for quick and easy calculations, making it an essential concept for students and professionals in fields such as engineering, architecture, and trigonometry. Additionally, the 30-60-90 triangle’s ability to generate similar triangles provides a powerful tool for solving complex geometric problems.

The 30-60-90 triangle is a testament to the elegance and simplicity of mathematics, and its study can deepen our understanding of the relationships between lengths and angles in triangles.

Whether you are a student or a teacher, taking the time to understand the 30-60-90 triangle will surely pay dividends in your future studies and work.