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The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If θ is the angle between two opposing sides in each quadrilateral, then the ratio of the two areas is given by sec 2 θ. For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e and f to give [2] [3]
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.
As a consequence of the Pythagorean theorem, the hypotenuse is the longest side of any right triangle; that is, the hypotenuse is longer than either of the triangle's legs. For example, given the length of the legs a = 5 and b = 12, then the sum of the legs squared is (5 × 5) + (12 × 12) = 169, the square of the hypotenuse.
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem .
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. [5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms ...
Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides. [1] [7] A kite can be constructed from the centers and crossing points of any two intersecting circles. [8] Kites as described here may be either convex or concave, although some sources restrict kite to mean only convex kites.
A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]