The 45 45 90 triangle calculator is a specialized tool to solve this special right triangle. You can find out what are the sides, hypotenuse, area, and perimeter of your shape by using the 45 45 90 triangle formula and learning about its ratio and rules.

If you are interested in knowing more about another popular right triangle, then you can check out 30 60 90 triangle tool and the calculator for special right triangles.

**How to solve a 45 45 90 triangle?**

The key to finding the formula for 45 45 90 triangle hypotenuse is to know the formula for 45 45 90 triangle hypotenuses.

The leg of a triangle when equal to a is calculated as follows:

- If the second leg equals a, then the total is equal to α
- the hypotenuse is equal to a√2
- where the area is equal to a2/2
- where the perimeter is equal to α (2 + 2)

Well, that’s a pretty easy question, but where does you get your perimeter from? The most popular methods for proving that equation is:

**By using Pythagoras’s Theorem:**

- You know the length of one leg a, so you know the length of the other leg as well, since both legs are equal.
- Use the Pythagorean theorem to find the hypotenuse:

a² + b² = c² => a² + a² = c² so c = √(2a²) = a√2

**Using Square Properties:**

Have you noticed that 45 45 90 is half a square, cut along the diagonal?

- We know that both legs equal a square
- It is probably not a surprise to you that the diagonal of the square is equal to side times square root of 2 – a√2. In our case, this diagonal equals the hypotenuse. What a quick calculation!

**Using of Trigonometry:**

You can use the properties of sine and cosine if you are familiar with trigonometry. It is a very special angle of 45°, so both of them are equal to √2/2. In other words:

- a/c = √2/2 so c = a√2

The basic formula for finding the area of a triangle is area = base * height / 2. There is a right angle between the leg we are using as a base and the leg we are using as a height, so in our case, one leg serves as a base. Hence, the area of a 454590 triangle is:

- `area = a² / 2`

To calculate the perimeter of the triangle, simply add all the 45 45 90 triangle sides together:

- perimeter = a + b + c = a + a + a√2 = α (2 + √2)

**Triangle Sides of 45 45 90:**

This triangle is formed by equal legs, so it can be quickly calculated from the equation c = a√2that the hypotenuse is the same as the legs in the triangle. In order to obtain the length of the side, we can use a = c√2/2 if the hypotenuse value is given.There are three triangles (four squares). In the red triangle is a 45 45 90 degrees angle triangle.

**Rules And Properties Of 45 45 90 Triangles:**

There is one right angle in this triangle, and two other angles are equal to 45 degrees. It implies that both sides – the legs – are equal in length and the hypotenuse can be calculated easily. The 45 45 90 triangles also have the following interesting properties:

- It is the only right triangle that is also an isosceles triangle
- It is the smallest in terms of the ratio of the hypotenuse to the legs
- It has the highest ratio of the altitude of the hypotenuse to the sum of the legs.

**Ratio of Triangle 45 45 90:**

The ratios of a 45 45 90 triangle are:

- 1: 1: 2 for angles (45°: 45°: 90°)
- 1: 1: √2 for sides (a: a: a√2)