What is the formula for Paythgoras

Pythagorean theorem

Probably the best-known theorem of geometry or mathematics is probably the Pythagorean theorem be:

In a right triangle, the sum of the leg squares is equal to the square of the hypotenuse.

Most people should use the Pythagorean theorem in the form of the equation a² + b² = c² be familiar, in this form it is easy to remember. The theorem establishes a relationship between the areas of the squares involved.

A solution for the theorem or application of the Pythagoras formula would be e.g .: 3² + 4² = 5². This is a very special solution because all numbers are integers. There are, by the way, an infinite number of such integer solutions of the Pythagorean theorem (5² + 12² = 13², 6² + 8² = 10², 7² + 24² = 25²), which are also called Pythagorean number triples.

Waterproof visualization of the Pythagorean theorem

The pouring of water shown in the following video clearly demonstrates the relationships. The content of the two smaller squares corresponds exactly to the content of the larger one.

Source: Wikipedia In a right triangle, the sum of the areas of the cathetus squares is equal to the area of ​​the hypotenuse square

Proof of the Pythagorean theorem

Evidence for this theorem is a dime a dozen, but not every student will be able to understand each of these proofs equally well. And not every proof / calculation method is suitable to bring the facts across easily and understandably. During my studies I came across a book that dealt exclusively with proofs of the Pythagorean theorem. There were a hundred of them. It doesn't have to be that much, the following 10 Pythagoras proofs (PDF) (Adrian Christen / Independent work as part of the lecture: Mathematics for Secondary School) should provide a good overview for the time being.

In my opinion, one of the easiest proofs to understand is the supplementary proof for the Pythagorean theorem. Here, the area of ​​the hypotenuse square is appropriately divided into two smaller squares and four right-angled triangles. And to go with it, there is also a musical variant including a corresponding proof. Have fun!

The video also points out the validity of the reverse of the Pythagorean theorem. If the formula a² + b² = c² applies to a triangle, i.e. the area of ​​the square above the hypotenuse corresponds to the area of ​​the squares of the cathetus, the right-angledness of the triangle follows directly from this. The educational and at the same time entertaining math song video comes from the DorFuchs.

In school life one comes across the theorems of Phytagoras, formulas that use him and proofs that use him again and again. The sentence is usually introduced in the 8th or 9th grade, but it appears again and again in many areas of mathematics in later school years.

For the general case, I would like to answer the question that is often asked whether the Pythagorean theorem can also be used to calculate the area of ​​a triangle. Of course, when I determine the perpendicularity of a triangle, I can deduce the area a · b / 2. You could also divide a non-right-angled triangle into two right-angled triangles by dropping a perpendicular - and then infer the area using the above argument. Nevertheless, this theorem is not a formula for calculating area.

Use cases for Pythagoras can be found not only in mathematical proofs, formulas, and calculations, but also in practical life. You just have to look. Bettermarks provides some examples under Applications to the Pythagorean Theorem.

Further theorems in mathematics, which revolve around calculations in right triangles, are the Euclid's cathetus theorem (a² = p · c or b² = q · c) and the Euclid's theorem of heights: h² = p · q. These sentences, too, only deal with the proportions of the side lengths, heights and projected sections.

To the person of Pythagoras of Samos:

Greek philosopher who lived around 570 BC. Is said to have been born on Samos and around 510 BC. Died in Metapont. At that time founded a very influential religious-philosophical movement. Incidentally, Pythagoras is also said to have been the first to believe that the earth was a sphere.

By Galilea at de.wikipedia [GFDL or CC-BY-SA-3.0], from Wikimedia Commons

Another well-known mathematical theorem, which also revolves around triangles and right angles, is the Thales theorem.

Image source: Wikipedia / Pythagoras