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The circumference is 2 π r, and the area of a triangle is half the base times the height, yielding the area π r 2 for the disk. Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , [ 2 ] but did not identify ...
Taking L to be the x-axis, the line integral between consecutive vertices (x i,y i) and (x i+1,y i+1) is given by the base times the mean height, namely (x i+1 − x i)(y i + y i+1)/2. The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal. The area of a triangle then ...
For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016 [7]) is 2π ⋅ 6371 2 | sin 90° − sin 66.56° | = 21.04 million km 2 (8.12 million sq mi), or 0.5 ⋅ | sin 90° − sin 66.56° | = 4.125% of the total surface area of the Earth ...
The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then = =.
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
Paul Nahin, a professor emeritus at the University of New Hampshire who wrote a book dedicated to Euler's formula and its applications in Fourier analysis, said Euler's identity is "of exquisite beauty". [8] Mathematics writer Constance Reid has said that Euler's identity is "the most famous formula in all mathematics". [9]
Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero. Brahmagupta's formula gives the area K {\displaystyle K} of a cyclic quadrilateral whose sides have lengths a , {\displaystyle a,} b , {\displaystyle b,} c , {\displaystyle c ...
Consider all cells (x, y) in which both x and y are integers between − r and r. Starting at 0, add 1 for each cell whose distance to the origin (0, 0) is less than or equal to r. When finished, divide the sum, representing the area of a circle of radius r, by r 2 to find the approximation of π. For example, if r is 5, then the cells ...