30 60 90 triangle with hypotenuse 1

There are many types of triangles. Acute, Obtuse, Isosceles, Equilateral triangles. Out of these different triangles, few of them are special. When we say special, it means the sides and angles which are predictable and consistent.

Thirty sixty ninety triangle

Out of all the other shortcuts, 30-60-90 is indeed a special Triangle.

What is a 30-60-90 Triangle?

It is a triangle where the angles are always 30, 60 and 90. As one angle is 90, so this triangle is always a right triangle. Thus, these angles form a right-angled triangle. Also, the sum of two acute angles is equal to the right angle, and these angles will be in the ratio 1 : 2 or 2 : 1.

Sides of a 30 60 90 Triangle

As explained above, it is a unique triangle with particular values of lengths and angles. Thus, the sides of 30 60 and 90 triangles are considered to be the Pythagorean triples. In general, the sides of a triangle with angles 30 degrees, 60 degrees and 90 degrees can be expressed as given in the below table:

Facts about the sides of 30 60 90 triangle:

• The side opposite to the angle 30° is always the shortest since 30 degrees is the smallest angle
• The side opposite to the angle 60° will be the medium length since 60 degrees is the mid-sized degree angle in this triangle
• The side opposite to the angle 90° will always be the largest since 90 degrees is the largest. This side of the triangle is called the hypotenuse

Area of 30 60 90 Triangle Formula

Consider the triangle of 30 60 90 in which the sides can be expressed as:

Here,

Base = x√3

Perpendicular (or Height) = x

Hypotenuse = 2x

We know that,

Area of triangle = (½) × Base × Height

= (½) × (x√3) × (x)

= (√3/2)x2

Example of 30 – 60 -90 rule

Example: Find the missing side of the given triangle.

Solution:

As it is a right triangle in which the hypotenuse is the double of one of the sides of the triangle. Thus, it is called a 30-60-90 triangle where a smaller angle will be 30. The longer side is always opposite to 60° and the missing side measures 3√3 units in the given figure.

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The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle.

The figure illustrates the ratio of the sides for the 30-60-90-degree triangle.

• Type 1: You know the short leg (the side across from the 30-degree angle). Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg.

• Type 2: You know the hypotenuse. Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg.

• Type 3: You know the long leg (the side across from the 60-degree angle). Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse.

  
  Finding the other sides of a 30-60-90 triangle when you know the hypotenuse

Because you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side: