Pythagoras Triangle Calculator

Our Pythagoras triangle calculator will explain to you the deep connection between the sides of a right triangle and help you solve every problem you may have involving them.

What are you going to find in this article?

• What is a Pythagoras triangle;
• The definition of the Pythagoras'theorem for right triangles;
• We will give you some proof of the Pythagoras theorem; and
• Teach you how to calculate the Pythagoras theorem in triangles with a right angle.

A Pythagoras triangle is nothing but a right triangle. But what is a right triangle? Right triangles are a special type of triangle where one of the angles has a value $90\degree$. You can learn more about them at our right triangle calculator!

It may not sound important, but that specific value confers to the shape many interesting properties. One of them concerns the sides of the triangle: the Pythagoras theorem.

The Pythagorean theorem in a triangle calculates the value of the third side given the values of two other sides.

It may sound similar to the SAS triangle (and somehow resembles it), but the steps we employ in the calculations are entirely different.

In any right triangle, if we identify:

• The perpendicular sides, called the catheti, $c_1$, and $c_2$;
• The slanted side, called the hypothenuse, $h$;

we can apply the following relationship:

This formula is the Pythagoras' theorem in a triangle with a right angle.

There are particular combinations of natural numbers which respect the Pythagorean theorem. We call them Pythagorean triples, and they are an interesting mathematical curiosity. Learn more about them at our Pythagorean triple calculator.

To calculate a Pythagoras triangle you need to first identify the elements you know, then apply one of the two versions of the Pythagorean theorem for a triangle to calculate the third element:

where $\text{i}$ and $\text{j}$ assumes the values $1$ and $2$ to complete the formula.

We can see an example of Pythagoras triangle calculations: say that we know:

• The hypotenuse, $h=6$; and
• A cathetus, $c_1=5$.

How do we calculate this Pythagoras triangle last side? Let's apply the second version of the theorem:

The number of proofs of the Pythagoras' theorem for triangles with a right angle is huge and still growing. One of the oldest, if not the oldest, proof uses the rearrangement of triangles and squares.

Draw a square with side $c_1+c_2$, and trace the right triangles with catheti $c_1$ and $c_2$. The hypotenuses of the four triangles identify a tilted square with side $h$.

Let's analyze the areas of the shapes:

• Each triangle has area $\tfrac{1}{2}c_1\cdot c_2$;
• The bigger square has area $(c_1+c_2)^2 = c_1^2+c_2^2+2\cdot c_1\cdot c_2$; and
• The smaller square has area $h^2$.

If we rearrange the triangles as in the following picture, we can draw a comparison.

As you can see, the four triangles are arranged in a way that creates two squares with sides, respectively, $c_1$ and $c_2$. Since the bigger square has the same size as before, we can compare the two figures:

If you erase the terms equal on both sides, you will find the Pythagorean's theorem:

There's another, less known, proof of the Pythagorean theorem by Garfield (the US president). Draw the following trapezoid:

And now calculate its area:

• As a sum of the triangles' areas; and
• As a trapezoid.

Since the height of the trapezoid is $(c_1+c_2)$, we can write:

Now define the areas of the triangles:

• The red triangle has area $\tfrac{1}{2}h^2$; and
• The yellow and blue triangle have area $\tfrac{1}{2}c_1\cdot c_2$.

Equal the two definition of the area of the trapezoid:

Expand the operations and multiply both sides by $2$:

Which returns, once again, the formula:

Do you want to learn more about the Pythagorean theorem? Check our pythagorean theorem calculator for a deeper analysis of this fundamental piece of geometry!