The Application of Pythagorean Theorem Calculator
The Pythagorean Theorem calculator is a special calculator that calculates the length of a hypotenuse or an unknown leg of a right triangle. The hypotenuse in the right triangle is the side that is opposite to the right angle. This side can be found using the hypotenuse formula, which is one more term for the Pythagorean Theorem used for solving the hypotenuse. As you remember the right triangle is a triangle with an angle that measure 90 degrees. The sum of the other two angles in the right triangle must also total 90 degrees, because the sum of the measures of all the angles in any triangle must be equal to 180 degrees. Below let’s learn more about the hypotenuse, the Pythagorean Theorem, and Pythagoras himself.
About Pythagoras
Pythagoras’s teachers were Germodamant and Pherecydes. Later, Pythagoras went to Miletus, where he met with other scientist – Thales. Thales advised him to go for the knowledge to Egypt and Pythagoras went there.
After being in Egypt for a while and studying the language and religion of the Egyptians, Pythagoras went to Memphis, where cunning priests offered him to pass a difficult test. Pythagoras did all the tests successfully. After learning everything that the priests could gave him, Pythagoras moved back home to Greece.
Later, Pythagoras decided to go on the overland journey, during which he was captured in captivity by Cambyses, king of Babylon. Babylonian mathematics was undoubtedly more developed (for example, Babylonians had a positional system of calculation) than Egyptian, and Pythagoras had something to learn from it as well.
After some time, Pythagoras fled to his homeland, Samos, where at the time the tyrant Polycrates reigned. Of course, Pythagoras did not want to be a servant, and he went to live in a cave in the vicinity of Samos. After months of claims by Polycrates, Pythagoras moved to Croton. In Croton, Pythagoras established a kind of religious and ethical secret brotherhood («Pythagoreans»), whose members pledged to conduct the so-called Pythagorean lifestyle. It was a religious association, a political club, and a scientific community. After 20 years, everybody knew about Pythagoreans.
Pythagoras and His Theorem
It’s hard to find someone who has never associated the name of Pythagoras with the Pythagorean Theorem. Probably even those people who only studied mathematics at high school still recall the so called «pants of Pythagoras» – the square constructed on the hypotenuse, which is equal to the two squares lying on the legs.
The reason why the Pythagorean Theorem is so popular is because it is simple, sophisticated, and important. This combination of two contradictory principles gives it a special power, making it very sophisticated.
In addition, the Pythagorean Theorem is of great importance: it is applied in the geometry almost with any function, and the fact that there are about 500 different proofs of this theorem (geometrical, algebraic, mechanical, etc.), shows a giant number among of its specific implementations.
The discovery of the Pythagorean Theorem is surrounded by a halo of legends. Proclus commented that Pythagoras in honor of his discovery can be found in various special tasks and drawings, as well as in the Egyptian triangle in the papyrus of the times of the pharaoh Amenemhat the first (2000 B.C.), and in Babylonian cuneiform tablets of the era of King Hammurabi (XVIII century B.C.), and in Old Indian geometrically theological treatise of the VII – V centuries B.C.
In the ancient Chinese treatise it is argued that in the XII century B.C. Chinese knew the properties of Egyptian triangle and the general notion of the theorem. Despite all this, the name of Pythagoras so strongly alloys with the Pythagorean Theorem, that now it is simply impossible to imagine that this phrase can fall apart. Today it is assumed that Pythagoras gave the first proof of the theorem, which bears his name. Alas, there are no traces left from this evidence.
The Notion of the Pythagorean Theorem
The Pythagorean Theorem is a relation used in Euclidean geometry related to the three sides of a right triangle. According to this relation, the sum of the squares of the sides of a right triangle is total the square of the hypotenuse. This theorem is often related to as the hypotenuse formula. If the sides of right triangle are a and b with the hypotenuse c, then the formula is a2+b2=c2. The formula can be calculated both at hand or with the help of Pythagorean Theorem calculator.
Even though, the theorem is credited to the ancient Greek mathematician and philosopher Pythagoras, there is no concrete evidence that Pythagoras himself was working on the theorem or proved it.
How to Calculate the Pythagorean Theorem
- Insert the lengths of the legs or hypotenuse in the formula. For example, assume you know that a=5, b=9 and you want to find out the length of the hypotenuse c.
- When you put the given values into the formula, you will get the following: 52+92=c2
- Square each value and you will get: 25+81=c2
- Combine the values and you will get 106= c2
- Take the square root of both equations’ sides and you will get: c=10.29. If you want to make sure the calculations are correct, you can double check it with the Pythagorean Theorem calculator.
The Pythagorean Theorem calculator will calculate the length of the hypotenuse c in the same manner. So if you don’t have a Pythagorean Theorem calculator at hand when it is needed, you will be able to calculate the length of the hypotenuse c by hand.
The Hypotenuse Formula
The hypotenuse formula is simply taking the Pythagorean Theorem and solving for the hypotenuse c. This formula can also be calculated on Pythagorean Theorem calculator. In order to solve for the hypotenuse c, take the square root of equation’s both legs a2+b2=c2and solve for c. When you do that, you will get the following: c=Vb2-a2. This is simply an extension of the Pythagorean Theorem and often is not associated with the name hypotenuse formula.
Types of Proofs of the Pythagorean Theorem
From ancient times mathematicians find more and more proofs of the Pythagorean Theorem, as well as more and more new designs of its evidence. There are more or less about five hundred proofs, but the pursuit of multiplying this number remains.
At this point, in the scientific literature there are 367 proofs of this theorem. This diversity can be explained only by the fundamental value of the theorem in geometry.
All of these proofs can be divided into various numbers of classes. The most famous are the method of proof areas and exotic axiomatic proof (for example, by means of differential equations). Let’s look at some of the most popular proofs of the theorem.
- The simplest proof.
The square built on the hypotenuse of a right triangle is equal to the sum of the squares constructed on the other two sides. The simplest proof of the theorem works if there is an isosceles right triangle. In fact, just a look at the number of isosceles right triangles verifies the validity of the theorem. For example, for the triangle ABC: a square built on the hypotenuse AC includes 4 original triangles, while squares built on sides include only two triangles. - Similitude method.
Among the algebraic proofs of the Pythagorean Theorem the method of similitude is the oldest and the most popular. Let’s assume there is a right triangle ABC with a right the right angle C. Draw the altitude from C and denote its base through H. The triangle ACH is similar to triangle ABC in two corners. Similarly, the triangle CBH is similar to triangle ABC. Introducing the notation BC=A, AC=B, AB=C. - Garfield Proof.
If from three right triangles we make a trapezoid, we can calculate the square of this figure using the formula for finding a square of a rectangular trapezoid, or as a sum of the squares of three triangles. - The proof of Euclid.
The idea of the proof of Euclid is the following: half of the square of a quadrate built on the hypotenuse is equal to the sum of half of the squares of the quadrates built on the sides, and then the square of the big quadrate and the square of two small quadrates are equal. - The proof of Leonardo da Vinci.
The main elements of this evidence are the symmetry and the movement. The square of the figure is the sum of half of the squares of the quadrates built on the sides, and the square of a given triangle. On the other hand, it is equal to half of the square of the quadrate built on the hypotenuse, plus the square of the original triangle.
The value of the theorem is that it is not only used in almost all modern technologies, but also opens the door to create new ones. The Pythagorean Theorem can help bring the majority of the theorems in geometry and solve many problems. Because of this, many scientists say the Pythagorean Theorem is geometry’s most important theorem. It is the foundation and the basis of all mathematical calculations and many various inventions.
It is very easy to solve various tasks in geometry with using the Pythagorean Theorem. In order to find the hypotenuse of the triangle you can use the Pythagorean Theorem Calculator or solve the task using the formula. Today, there are many mathematical websites that offer free Pythagorean Theorem Calculator that help you find the length of a hypotenuse.
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