45-45-90 Triangle

A 45-45-90 triangle is a special right triangle. The other type of special right triangle is 30-60-90. These numbers represent the degree measures of the angles.


The reason these triangles are considered special is because of the ratios of their sides - they are always the same!

Special right triangles hold many applications in both geometry and trigonometry.

In this lesson you will learn the general formula for the ratios, and how to find missing sides of any 30 60 90 right triangle. We even offer a free study guide that easy explains the steps easily on one page!

The example below shows the basic ratios of the triangle. Regardless of what the "x" value is, the ratios will always hold true.

The General Formula of

a 45-45-90 Triangle


Also Known as an Isosceles Right Triangle

Because both of the base angles are equal to 45°, the converse of the isosceles triangle theorem tells us that both of the legs are equal.

Therefore, we will refer to the two congruent sides as legs, and the other side (which is opposite the right angle) as the hypotenuse.

More information about the hypotenuse from the pythagorean theorem.


How to Find the Missing Side of the Triangle

You will always be given one of the three sides. What you do from there depends on which side that you are given. Consider both cases below.

Case #1: I know one of the legs

  • The other leg is equal.
  • Find Hyp: Multiply by Sqrt(2)

Case #2: I know the Hypotenuse

  • Find Legs: Divide by Sqrt(2)

Download GradeA's conveinent .pdf file: special right triangle study guide.


Examples: Both Cases

Case #1: I know a Leg

Case #2: I know the Hypotenuse

Please notice that in both cases you are either multipying by the square root of 2 or you are dividing by the square root of 2. When you divide by a square root, don't forget to rationalize the denominator by multiplying both the top and the bottom of the fraction by the radical. Good luck!

Additional Information

Learn about the other special right triangle the 30-60-90 or you may also be interested in other geometry formulas.

Return from 45-45-90 triangle to free geometry help.

Visit the GradeA homepage for more free math help.

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