45 - 45 - 90 Triangle Rules and How to Use Them

Introduction to 45 - 45 - 90 Triangles

A 45° - 45° - 90° triangle is an isosceles triangle ◣ and can also be approached as a square cut in half. As the name suggests, the triangle has two 45° angles and one 90° angle. This makes the two sides equal.

How To Find the Missing Sides in a 45 - 45 - 90 Triangle

In a 45° - 45° - 90° triangle, the sides opposite to the 45° angles are equal in length and the hypotenuse, the side opposing the 90° angle, is the longest side. Its length can be determined by multiplying the length of a leg by the square root of 2, as shown in the image below:

Now, let’s practice doing calculations with some 45° - 45° - 90° Triangles! 💪

Examples of 45 - 45 - 90 Triangles

1. Find the lengths of a and b in the following 45° - 45° - 90° triangle. 

Answer: As you know, the sides opposing the 45° degree angles are equal to each other, so “a” will be equal to 2. On the other hand, “b”, will take a bit more work. To find that length, multiply the length of “a”, aka 2, by the square root of 2. 

So “b” equals 2(√2). Most teachers accept this as a simplified answer, but if not, just plug 2(√2) into the calculator 💻 and get your decimal answer there. 

What happens if you are only given the length of the hypotenuse? You are going to have to work ← backward. So instead of multiplying by √2, you’ll have to divide by √2. Let’s see what this looks like:

2. Find the length of x and k in this 45° - 45° - 90° triangle. 

We got 16, just like in the original question! 

Let’s try another problem.

3. Find the length of x and y in this 45° - 45° - 90° triangle. 

And the answer is 21, just like in the original question! 

Variations of 45 - 45 - 90 Triangles

Let’s look at a few different types of variations you may see regarding the 45° - 45° - 90° triangle. 

1. Can you determine if this is a 45° - 45° - 90° triangle?

Answer: You can determine that this is a 45° - 45° - 90° triangle. We know this because there is a 90° angle. The 90° angle is represented by the little square in the corner. Since the angles inside of a triangle add up to 180° we know that if one angle equals 90°, then the other two angles have to add up to 90°. We also notice that the same symbol represents both missing angles, suggesting that they must be equal. Since you know that the missing angles add up to 90°, here you can just divide 90° by two, and get 45°, confirming that this is a 45° - 45° - 90° triangle. 

45° - 45° - 90° triangles may also look like this: 

Disclaimer: *  Don’t ever look at a triangle and assume it’s a 45° - 45° - 90° triangle just because the two sides look like they are the same length. Most of these triangles aren’t drawn to scale. * 🧠

You can also use the Pythagorean Theorem to solve a 45° - 45° - 90° triangle. If you need a refresher on the Pythagorean Theorem, check out this article from Fiveable. 

Conclusion 

45° - 45° - 90° triangles are very fun and relatively easy to solve. Have fun trying to solve them! For more Fiveable math content, check out this article or this article!