Special Triangle: Rules & Formulas

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  • 0:04 Special Right Triangles
  • 1:14 The 45-45-90 Triangle
  • 1:48 The 30-60-90 Triangle
  • 2:26 Some Examples
  • 3:42 Lesson Summary
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Lesson Transcript
Instructor: Vanessa Botts
Finding the missing side of a triangle is pretty simple when that triangle is 'special'. In this lesson, you will learn about two types of special right triangles, their properties and shortcuts to find the missing side length.

Special Right Triangles

All right triangles have special properties, but there are certain ones that have some features that make it easier to calculate the length of a missing side without using the Pythagorean theorem or trigonometric functions. This is why they are called special triangles.

First, it is important to remember that a right triangle is a triangle with one 90-degree angle. The longest side is always the hypotenuse, and it will always be located across from the 90-degree angle. The other two sides, which we call the legs, may or may not be of equal length.

right triangle

Now that we've reviewed what a right triangle looks like, we can begin to talk about the two most studied types.

  • The first one is a 45-45-90 triangle
  • The second one is a 30-60-90 triangle

A 45-45-90 triangle is a special right triangle whose angles are 45, 45, and 90 degrees. The 30-60-90 triangle is a right triangle whose angles are 30, 60 and 90 degrees. Both of these special triangles have something in common: their sides have a particular ratio.

Now, we will explore each type in more detail.

The 45-45-90 Triangle

45 traingle image 1

The lengths of the sides are in the ratio of:

triangle ratio

Based on the 1:1 ratio of the two sides, we can see that its two legs, are the same length. We will designate each as L. The hypotenuse, H, will then be the product of L and the square root of 2. To make computations simple, we can use formulas that look like this:

45 formula

45 formula for leg

The 30-60-90 Triangle

30 60 90 Triangle

The lengths of the sides triangle are in the ratio of:


For this triangle we will designate the shortest leg as S and the medium leg as L. Remember that the hypotenuse is always the longest side, and we will call it H. The formulas to use for this type of triangle are:

30 formula 1

30 form 3

30 form 2

Now it's time to put what we learned into practice. In the following examples we will see how useful it is to know the triangles' properties and ratios.

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