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.

What is a Pythagoras triangle?

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°90\degree.

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.

What is the Pythagorean theorem: right triangles and fellow squares

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, c1c_1, and c2c_2;
  • The slanted side, called the hypothenuse, hh;

we can apply the following relationship:

h=c12+c22h=\sqrt{c_1^2+c_2^2}

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.

How do I calculate a Pythagoras triangle?

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:

h=c12+c22orci=h2cj2\begin{align*} h&=\sqrt{c_1^2+c_2^2}\\ &\text{or}\\ c_\text{i}&=\sqrt{h^2-c_\text{j}^2} \end{align*}

Where i\text{i} and j\text{j} assumes the values 11 and 22 to complete the formula.

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

  • The hypotenuse, h=6h=6; and
  • A cathetus, c1=5c_1=5.

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

c2=h2c12c2=6252=3625=113.32\begin{align*} c_{2}&=\sqrt{h^2-c_{1}^2}\\ c_2&=\sqrt{6^2-5^2} =\sqrt{36-25}\\ &=\sqrt{11}\simeq 3.32 \end{align*}

Proofs of the Pythagoras' theorem in a triangle

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 c1+c2c_1+c_2, and trace the right triangles with catheti c1c_1 and c2c_2. The hypotenuses of the four triangles identify a tilted square with side hh.

Rearrangement proof of pythagorean theorem 1
Start by analyzing this figure.

Let's analyze the areas of the shapes:

  • Each triangle has area 12c1c2\tfrac{1}{2}c_1\cdot c_2;
  • The bigger square has area (c1+c2)2=c12+c22+2c1c2(c_1+c_2)^2 = c_1^2+c_2^2+2\cdot c_1\cdot c_2; and
  • The smaller square has area h2h^2.

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

Rearrangement proof of pythagorean theorem 2
If you rearrange the shapes you can easily prove the Pythagorean theorem.

As you can see, the four triangles are arranged in a way that creates two squares with sides, respectively, c1c_1 and c2c_2. Since the bigger square has the same size as before, we can compare the two figures:

4(12c1c2)+h2=4(12c1c2)+c12+c22\begin{align*} 4\cdot &\left(\frac{1}{2} c_1\cdot c_2 \right)+ h^2 = \\ \\4\cdot&\left(\frac{1}{2}c_1\cdot c_2\right) + c_1^2 + c_2^2 \end{align*}

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

h2=c12+c22 h^2 = c_1^2 + c_2^2

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 (c1+c2)(c_1+c_2), we can write:

A=(c1+c2)(c1+c2)2A=\frac{(c_1+c_2)\cdot(c_1+c_2)}{2}
Garfield's Pythagorean theorem proof
The sketch for the Garfield's proof of the Pythagorean theorem.

Now define the areas of the triangles:

  • The red triangle has area 12h2\tfrac{1}{2}h^2; and
  • The yellow and blue triangle have area 12c1c2\tfrac{1}{2}c_1\cdot c_2.

Equal the two definition of the area of the trapezoid:

12(c1+c2)(c1+c2)=12h2+2(12c1c2)\begin{align*} &\frac{1}{2}(c_1+c_2)\cdot(c_1+c_2)=\\ &\frac{1}{2}h^2+2\left(\frac{1}{2}c_1\cdot c_2\right) \end{align*}

Expand the operations and multiply both sides by 22:

c12+c22+2c1c2=h2+2c1c2\begin{align*} &c_1^2+c_2^2+2\cdot c_1\cdot c_2=\\ &h^2+2\cdot c_1\cdot c_2 \end{align*}

Which returns, once again, the formula:

h2=c12+c22h^2 = c_1^2 + c_2^2

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!

FAQ

What is a Pythagoras triangle?

A Pythagoras triangle is any right triangle. Pythagoras triangle satisfies the Pythagorean theorem, which allows calculating the length of the third side of an angle knowing the value of the other two. The Pyhtagoras theorem in a triangle states that:
h² = c₁² + c₂²
Where:

  • h is the hypotenuse; and
  • c₁ and c₂ are the catheti.

Is a right triangle with sides 3, 4, and 5 a Pythagoras triangle?

A right triangle with catheti c₁ = 3 and c_2 = 4, and hypotenuse h = 5 is an example of Pythagoras triangle, since it fulfils the equation:
h² = c₁² + c₂²
Moreover, this triangle is an example of a Pythagorean triple since all its sides are natural numbers.

How do I calculate a triangle with the Pythagorean theorem?

To calculate a Pythagoras triangle, follow these steps:

  1. Identify the known elements.
  2. Find the third value:
    *If you know both catheti, apply the following formula:
    h² = c₁² + c₂²
    • If you know a cathetus and the hypotenuse, apply the formula:
      cᵢ² = h² + cⱼ²

What is the third side of a right triangle with catheti 4 and 6?

The third side is 7.21. To calculate the hypotenuse of this triangle, apply the following equation:
h² = c₁² + c₂²
and substitute the values of c₁ and c₂:
h = sqrt(4² + 6²) = sqrt(16 + 36) = sqrt(52) = 7.21

Davide Borchia
A Pythagorean triangle with all sides marked
a² + b² = c²
a
cm
b
cm
c
cm
Area
cm²
Perimeter
cm
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