45-45-90 Right Triangles

A 45-45-90 triangle is a special type of right-angled triangle where the two non-right angles are both 45 degrees. This also makes it an isosceles triangle, meaning that the two sides opposite these equal angles (the legs) are of equal length. This unique angle combination leads to some important properties and relationships between the sides of the triangle.

Recognizing special right triangles in geometry can provide a shortcut when answering some questions. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem formula, but if you can see that it is a special triangle it can save you some calculations.

What is a 45-45-90 Triangle?
A 45-45-90 triangle is a special right triangle whose angles are 45°, 45° and 90°. The lengths of the sides of a 45-45-90 triangle are in the ratio of 1:1:√2.

leg : leg : hypotenuse = 1 : 1 : √2

The following diagram shows a 45-45-90 triangle and the ratio of its sides. Scroll down the page for more examples and solutions using the 45-45-90 triangle.

Note:

• A 45-45-90 triangle is an isosceles right triangle. It is also sometimes called a 45-45 right triangle.
  
• Any right triangle with two sides of equal lengths would be a 45-45-90 triangle.
• You can also recognize a 45-45-90 triangle by the angles. As long as you know that one of the angles in the right-angle triangle is 45° then it must be a 45-45-90 special right triangle.
• A right triangle with a 45° angle must be a 45-45-90 special right triangle.

How to Solve Problems with 45-45-90 Triangles (Find Missing Sides):
You can use the side length relationships to find the missing sides of a 45-45-90 triangle if you know the length of one side:

1. If you know the length of one leg (x):
   

• The other leg has the same length (x).
  
• The hypotenuse is the length of a leg multiplied by \( \sqrt{2} \)
  

2. If you know the length of the hypotenuse (h):
   

• The length of each leg is the length of the hypotenuse divided by \( \sqrt{2} \)
  

How to solve problems with 45-45-90 triangles?

Example 1:
Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches.

Solution:
Step 1: This is a right triangle with two equal sides so it must be a 45-45-90 triangle.

Step 2: You are given that the both the sides are 3. If the first and second value of the ratio n:n:n√2 is 3 then the length of the third side is 3√2

Answer: The length of the hypotenuse is 3√2 inches.

Example 2:
Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 4√2 inches and one of the angles is 45°.

Solution:
Step 1: This is a right triangle with a 45° so it must be a 45-45-90 triangle.

Step 2: You are given that the hypotenuse is 4√2. If the third value of the ratio n:n:n√2 is 4√2 then the lengths of the other two sides must 4.

Answer: The lengths of the two sides are both 4 inches.

Videos
The following videos show more examples of 45-45-90 triangles.

How to find the length of a leg or hypotenuse in a 45-45-90 triangle using the Pythagorean Theorem and then derive the ratio between the length of a leg and the hypotenuse?
This video gives an introduction to the 45-45-90 triangles and shows how to derive the ratio between the lengths of legs and the hypotenuse.

• Show Video Lesson

How to solve a 45-45-90 triangle given the length of one side by using the ratio?

• Show Video Lesson

Special Right Triangles in Geometry: 45-45-90 and 30-60-90
This video discusses two special right triangles, how to derive the formulas to find the lengths of the sides of the triangles by knowing the length of one side, and then does a few examples using them.

• Show Video Lesson

Example problems of finding the sides of a 45-45-90 triangle with answer in simplest radical form.

• Show Video Lesson

Why is this important?
Knowing the properties of 45-45-90 triangles allows you to quickly solve for missing side lengths without having to use the Pythagorean theorem every time. This is especially useful in geometry, trigonometry, and various applications involving right triangles.

Common Mistakes to Avoid
❌ Assuming all sides are equal: Only the legs are equal; the hypotenuse is longer.
❌ Forgetting √2: The hypotenuse is always Leg × √2
❌ Misapplying ratios: Only works for 45-45-90 triangles (not 30-60-90).

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