What is the length of a leg of a 45 45 90 triangle if the hypotenuse measures 12 cm?

A 45 45 90 triangle is a special right triangle because you can use short cuts to find leg length and hypotenuse length. This video solves two problems involving leg length and hypotenuse length.

Highlights of the video

A 45-45-90 triangle is an isosceles triangle, which means two sides are the same, has a right angle.

The angles, the two acute angles are two 45 degree angles, and are congruent.

You can solve a 45 45 90 triangle with one side because there are some special rules.

Let's look at the rules for 45- 45 -90. If you know one leg you know the other leg because these are isosceles, and the two sides are the same.

There is a ratio that all you need to do is take one of the legs and multiple by the square root of two to get the length of the hypotenuse.

The two legs of a 45 45 90 are the same, and two times the square root two gives us the length of the hypotenuse.

If we know this leg is 7 then the other leg is 7 because it is isosceles and the ratio says I just multiple this 7 times the square root of 2 to get the hypotenuse.

What if you don't know the two legs but you do know the hypotenuse.

Since the rule is you take the leg times square root 2 to go from the leg to the hypotenuse, to get the hypotenuse just reverse the operation and divide by the square root of two.

I will take 12 and divide it by the square root of two. You can't have a square root on the bottom of a fraction so I'm going to rationalize that by the factor of one so I have square root of two over square root of two, that's really just one.

The square root of two times square root of two is equal to the square root of four, which is two. Now on the top I have twelve times the square root of two.

I can't multiple these together so I just stick them next to each other ,and then let's divide these outside coefficients, which is 12 divided by 2, which is 6 and that simplifies to 6 root two.

So that means each leg is 6 square root two.

In summary, to get hypotenuse you multiple by square root of two and to get leg length you divide by the square of two.

Properties of 45 45 90 special right triangles, video, and an applet.The

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45 45 90 triangle calculator is a dedicated tool to solve this special right triangle. Find out what are the sides, hypotenuse, area and perimeter of your shape and learn about 45 45 90 triangle formula, ratio and rules.

If you want to know more about another popular right triangles, check out this 30 60 90 triangle tool and the calculator for special right triangles.

If you are wondering how to find the formula for 45 45 90 triangle hypotenuse, you're at a right place.
If the leg of the triangle is equal to a, then:

the second leg is also equal to a
the hypotenuse is a√2
the area is equal to a²/2
the perimeter equals a(2 + √2)

OK, looks easy, but where does it come from? There are a couple of methods to prove that equation, the most popular between them are:

Using the Pythagorean theorem

As you know one leg length a, you the know the length of the other as well, as both of them are equal.
Find the hypotenuse from the Pythagorean theorem:

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

Using the properties of the square

Did you notice that the 45 45 90 triangle is half of a square, cut along the square's diagonal?

Again, we know that both legs are equal to a
As you probably remember, the diagonal of the square is equal to side times square root of 2 - a√2. In our case, this diagonal is equal to the hypotenuse. That was quick!

If you heard about trigonometry, you could use the properties of sine and cosine. For this special angle of 45°, both of them are equal to √2/2. So:
a/c = √2/2 so c = a√2

To find the area of such triangle, use the basic triangle area formula is area = base * height / 2. In our case, one leg is a base and the other is the height, as there is a right angle between them. So the area of 45 45 90 triangles is:

`area = a² / 2`

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

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

The legs of such a triangle are equal, the hypotenuse is calculated immediately from the equation c = a√2. If the hypotenuse value is given, the side length will be equal to a = c√2/2.

Triangles (set squares). The red one is the 45 45 90 degree angle triangle

The most important rule is that this triangle has one right angle, and two other angles are equal to 45°. It implies that two sides - legs - are equal in length and the hypotenuse can be easily calculated. The other interesting properties of the 45 45 90 triangles are:

It's the only possible right triangle that is also an isosceles triangle
It has the smallest ratio of the hypotenuse to the sum of the legs
It has the greatest ratio of the altitude from the hypotenuse to the sum of the legs

In a 45 45 90 triangle, the ratios are equal to:

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

Have a look at this real-life example to catch on the 45 45 90 triangle rules.

Assume we want to solve the isosceles triangle from a triangle set.

Type the given value. In our case, the easiest way is to type the length of the part with the scale. The usual leg length is 9 inches, so type that value into a or b box.
The 45 45 90 triangle calculator shows the remaining parameters. Now you know:

hypotenuse length - 9 in * √2 = 12.73 in
area - 9 in * 9 in / 2 = 40.5 in²
perimeter - 9 in + 9 in + 9 in * √2 = 30.73 in

Remember that every time you can change the units displayed by simply clicking on the unit name. Also, don't forget that our calculator is a flexible tool - if you only know the area, the hypotenuse or even the perimeter, it can calculate the remaining parameters as well. Awesome!