30-60-90 Triangles

This will show you the concept of a 30-60-90 triangle which includes the relationship which occurs between the angles and the sides in them. There are many types and different names of triangles.

If you have any experience with geometry, then you know that there are different types of triangles which are divided by the length of sides that is isosceles, equivalent or scalene or by the angles of triangles that is right, acute and obtuse. There is a special kind of triangle, which is a 30-60-90 triangle, which is a right angled triangle with two acute angles which are 30 and 60 degrees.

A 30-60-90 triangle is special due to the relationships of sides. Hopefully you know the hypotenuse in the right angle which is the longest side across the right angles. It turns in to a 30-60-90 triangle where you can find out the measure of any three sides just by knowing the measure of just one side of the triangle.

The hypotenuse is the same as twice the length of the short legs which is the side of the 30 degrees. The longer leg is across the angle of 60 degree which is the same as the multiplication of the shorter leg by the square root of 3.

Note that the short side always acts as a bridge between the other two sides of the triangle. You can get the longer leg to the hypotenuse but if you pass through the short leg by just evaluating its value. There is no direct path from the long leg to the hypotenuse.

To obtain the hypotenuse (2a), multiply the shortest length (a) by 2.

To obtain the shortest leg (a), do either of the following:

  • divide the other leg (a√3) by √3
  • divide the hypotenuse (2a) by 2

To obtain the other leg, multiply the shortest leg (a) by √3.


Example 1 – find out the other two lengths of the triangle where the shortest side is 10 of 30-60-90 triangle.

triangles 1

Thus, the given side is 10 across the angle of 30 degrees, which is the shorter side. For finding out the long leg you just need to multiply it by the square root of 3 to get 10√3. For finding the hypotenuse you need to multiply the short leg by 2. So, we will get 10 * 2 = 20.

Example 2 – Find out the length of the other two sides of the 30-60-90 triangle with the hypotenuse being 15.

triangles 2

Therefore, as the given length of the hypotenuse is 15 where we need to find out the short leg. To find out the short side, we need to just divide the hypotenuse by 2 which is 15/2 = 7.5. Now we know the value of the short side, for finding the long side, we just need to multiply it by the root of 3 which is 7.5√3.

Thus, here we have a clear idea of the concept of the right-angled triangle with 30, 60 and 90 degrees. For finding the hypotenuse, you just need to multiply the value by 2, and to find out the value of short leg, just divide it by 2 and if there is a need to find out the value of long side, then just multiply by the square root of 3.