In mathematics, the Pythagorean Theorem or Pythagoras' Theorem is a relation in Euclidean geometry among the three sides of a right triangle, right-angled triangle. In terms of areas, it states:
In any right triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
C represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagorean Theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they fitted it into a mathematical framework.
The theorem is about both areas and lengths, or can be said to have both areal and metric interpretations. Some proofs of the theorem are based on one interpretation, some upon the other, using both algebraic and geometric techniques. The theorem can be generalized in various ways, including higher dimensional spaces, to spaces that are not Euclidean, to objects that are not right angle triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.
As stated above, if c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, Pythagoras' theorem can be expressed as the Pythagorean equation:
If the length of both a and b are known, then c can be calculated as follows:
If the length of hypotenuse c and one leg a or b are known, the length of the other leg can be calculated with the following equations:
Or
The Pythagorean equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle, the law of cosines reduces to the Pythagorean equation.
In any right triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
C represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagorean Theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they fitted it into a mathematical framework.
The theorem is about both areas and lengths, or can be said to have both areal and metric interpretations. Some proofs of the theorem are based on one interpretation, some upon the other, using both algebraic and geometric techniques. The theorem can be generalized in various ways, including higher dimensional spaces, to spaces that are not Euclidean, to objects that are not right angle triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.
As stated above, if c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, Pythagoras' theorem can be expressed as the Pythagorean equation:
If the length of both a and b are known, then c can be calculated as follows:
If the length of hypotenuse c and one leg a or b are known, the length of the other leg can be calculated with the following equations:
Or
The Pythagorean equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle, the law of cosines reduces to the Pythagorean equation.
