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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.
A right triangle with the hypotenuse c. In a right triangle, the hypotenuse is the side that is opposite the right angle, while the other two sides are called the catheti or legs. [7] The length of the hypotenuse can be calculated using the square root function implied by the Pythagorean theorem.
The legs and hypotenuse of a right triangle satisfy the Pythagorean theorem: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, + =. If the lengths of all three sides of a right triangle are integers, the triangle is called a Pythagorean triangle and its side lengths are collectively known as a ...
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae , it is one of the basic relations between the sine and cosine functions.
The Pythagorean theorem: 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). Although Pythagoras is most famous today for his alleged mathematical discoveries, [132] [207] classical historians dispute whether he himself ever actually made any significant contributions to the field.
Set square shaped as 45° - 45° - 90° triangle The side lengths of a 45° - 45° - 90° triangle 45° - 45° - 90° right triangle of hypotenuse length 1.. In plane geometry, dividing a square along its diagonal results in two isosceles right triangles, each with one right angle (90°, π / 2 radians) and two other congruent angles each measuring half of a right angle (45°, or ...
This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from to as its hypotenuse. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the ...
By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse. The term leg , in addition to referring to a cathetus of a right triangle, is also used to refer to either of the equal sides of an isosceles triangle or to either of the non-parallel sides of a trapezoid .