What is the length of a leg of an isosceles right triangle if the hypotenuse is 8 √ 2 cm long?

If the square of the hypotenuse of an isosceles right triangle is 128 cm2, find the length of each side.

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First note that every isosceles right triangle is similar. They all have the same shape, they're just scaled differently.

Now consider the isosceles right triangle with the two equal sides being $1$ unit. These have to be the mutually perpendicular sides (technically called the catheti) because the hypotenuse is strictly longer than any other side. Apply Pythagoras' Theorem. The hypotenuse $h$ is given by $h^2 = 1^2 + 1^2$, giving $h = \sqrt 2$.

You now know the ratios of the sides of every isosceles right triangle, i.e. $1:1:\sqrt 2$.

To apply it, if you're given that the hypotenuse is $13$, then the other two sides are each $\frac{13}{\sqrt 2} $. This is often rearranged to $\frac{13\sqrt 2}{2} $ to avoid an irrational number in the denominator.

Isosceles triangle calculator is the best choice if you are looking for a quick solution to your geometry problems. Find out the isosceles triangle area, its perimeter, inradius, circumradius, heights and angles - all in one place. If you want to build a kennel, find out the area of Greek temple isosceles pediment or simply do your maths homework, this tool is here for you. Experiment with the calculator or keep reading to find out more about the isosceles triangle formulas.

An isosceles triangle is a triangle with two sides of equal length, which are called legs. The third side of the triangle is called base. Vertex angle is the angle between the legs and the angles with the base as one of their sides are called the base angles.

Here are the most important properties of isosceles triangles:

• It has an axis of symmetry along its vertex height;
• Two angles opposite the legs are equal in length; and
• The isosceles triangle can be acute, right, or obtuse, but it depends only on the vertex angle (base angles are always acute)

The equilateral triangle is a special case of an isosceles triangle. You can learn about all the possible types of triangles in the classifying triangles calculator.

To calculate the isosceles triangle area, you can use many different formulas. The most popular ones are the equations:

1. Given leg a and base b:

   area = (1/4) × b × √( 4 × a² - b² )

2. Given h height from apex and base b or h2 height from other two vertices and leg a:

   area = 0.5 × h × b = 0.5 × h2 × a

3. Given any angle and leg or base

   area = (1/2) × a × b × sin(base_angle) = (1/2) × a² × sin(vertex_angle)

Also, you can check our triangle area calculator to find out other equations, which work for every type of triangle, not only for the isosceles one.

To calculate the isosceles triangle perimeter, simply add all the triangle sides:
perimeter = a + a + b = 2 × a + b

Isosceles triangle theorem, also known as the base angles theorem, claims that if two sides of a triangle are congruent, then the angles opposite to these sides are congruent.

Also, the converse theorem exists, stating that if two angles of a triangle are congruent, then the sides opposite those angles are congruent.

A golden triangle, which is also called sublime triangle is an isosceles triangle in which the leg is in the golden ratio to the base:

a / b = φ ~ 1.618

The golden triangle has some unusual properties:

• It's the only triangle with three angles in 2:2:1 proportions
• It's the shape of the triangles found in the points of pentagrams
• It's used to form a logarithmic spiral

Let's find out how to use this tool on a simple example. Have a look at this step-by-step solution:

1. Determine what is your first given value. Assume we want to check the properties of the golden triangle. Type 1.681 inches into leg box.
2. Enter second known parameter. For example, take a base equal to 1 in.
3. All the other parameters are calculated in the blink of an eye! We checked for instance that isosceles triangle perimeter is 4.236 in and that the angles in the golden triangle are equal to 72° and 36° - the ratio is equal to 2:2:1, indeed.

You can use this calculator to determine different parameters than in the example, but remember that there are in general two distinct isosceles triangles with given area and other parameter, e.g. leg length. Our calculator will show one possible solution.

To compute the area of an isosceles triangle with leg a and base b, follow these steps:

1. Apply the Pythagorean theorem to find the height: √( a² - b²/4 ).

2. Apply the standard triangle area formula, i.e., multiply base b by the height found in Step 1 and then divide by 2.

3. That is, the final formula we got is:

   area = ½ × b × √( a² - b²/4 )

We compute the perimeter of an isosceles triangle with leg a and base b with the help of the formula perimeter = 2 × a + b. This formula makes use of the fact that the two legs of an isosceles triangle are of equal length.

The answer is 6.93. To derive it, we can use the formula area = ½ × b × √( a² - b²/4 ) with a = b = 4.

Alternatively, we can notice that, in fact, we have here an equilateral triangle: the are formula simplifies to area = a² × √3 / 4 with a = 4.