# Pythagoras theorem wikipedia

Consider the triangle shown below. This figure is clearly a squaresince all the angles are right anglesand the lines connecting the corners are easily seen to be straight. Now to calculate the area *pythagoras theorem wikipedia* this figure.

In mathematics , the Pythagorean theorem or Pythagoras's theorem is a statement about the sides of a right triangle. One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs.

## Pythagoras theorem wikipedia

His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato , Aristotle , and, through them, the West in general. Knowledge of his life is clouded by legend. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included aspects of vegetarianism. The teaching most securely identified with Pythagoras is metempsychosis , or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis , which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton. Following Croton's decisive victory over Sybaris in around BC, Pythagoras's followers came into conflict with supporters of democracy , and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or he may have escaped to Metapontum and died there. In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem , Pythagorean tuning , the five regular solids , the Theory of Proportions , the sphericity of the Earth , and the identity of the morning and evening stars as the planet Venus.

Main article: Tree of Pythagorean triples. Page Talk.

In mathematics , the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a , b and the hypotenuse c , sometimes called the Pythagorean equation : [1]. The theorem is named for the Greek philosopher Pythagoras , born around BC. The theorem has been proved numerous times by many different methods — possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, [2] explaining its name. When used for this, it is also known as a hopscotch pattern [3] or pinwheel pattern , [4] but it should not be confused with the mathematical pinwheel tiling , an unrelated pattern. This tiling has four-way rotational symmetry around each of its squares. When the ratio of the side lengths of the two squares is an irrational number such as the golden ratio , its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions have also been studied. The Pythagorean tiling is the unique tiling by squares of two different sizes that is both unilateral no two squares have a common side and equitransitive each two squares of the same size can be mapped into each other by a symmetry of the tiling. Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagons.

### Pythagoras theorem wikipedia

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Cicero, De re publica 2, 28— Article Talk. The translations also leave the area unchanged, as they do not alter the shapes at all. The sum of the areas of the two squares on the legs a and b equals the area of the square on the hypotenuse c. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements , and that the theory of proportions needed further development at that time. Ann Arbor, Michigan: Edwards Brothers. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse — or conversely the large square can be divided as shown into pieces that fill the other two. During Pythagoras's formative years, Samos was a thriving cultural hub known for its feats of advanced architectural engineering, including the building of the Tunnel of Eupalinos , and for its riotous festival culture. Other ancient writers, however, claimed that Pythagoras had learned these teachings from the Magi in Persia or even from Zoroaster himself. Sign up to read all wikis and quizzes in math, science, and engineering topics. The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c , as shown in the lower part of the diagram. The wrestler Milo of Croton was said to have been a close associate of Pythagoras [95] and was credited with having saved the philosopher's life when a roof was about to collapse. Dear Wikiwand AI, let's keep it short by simply answering these key questions:.

In a right triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides. The Khan Academy has video material relating to this topic which you may find easier to follow:.

This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:. One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable so the ratio of which is not a rational number can be constructed using a straightedge and compass. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit. Pythagoras of Samos was a famous Greek mathematician and philosopher c. In the presentation above, it is said that all Pythagorean triples are uniquely obtained from Euclid's formula "after the exchange of a and b , if a is even". The underlying question is why Euclid did not use this proof, but invented another. Geometrically r is the distance of the z from zero or the origin O in the complex plane. In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. The theorem has been proved numerous times by many different methods — possibly the most for any mathematical theorem.

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