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The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.
A circle of radius 5 centered at the origin has area 25 ... For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 ...
A circle of radius 1 (using this distance) is the von Neumann neighborhood of its centre. A circle of radius r for the Chebyshev distance (L ∞ metric) on a plane is also a square with side length 2r parallel to the
A page from Archimedes' Measurement of a Circle. Measurement of a Circle or Dimension of the Circle (Greek: Κύκλου μέτρησις, Kuklou metrēsis) [1] is a treatise that consists of three propositions, probably made by Archimedes, ca. 250 BCE. [2] [3] The treatise is only a fraction of what was a longer work. [4] [5]
where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width. = where A is the area of a circle. More generally, =
The area within a circle is equal to the radius multiplied by half the circumference, or A = r x C /2 = r x r x π.. Liu Hui argued: "Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the ...
What can be stated is that as the central angle gets smaller (or alternately the radius gets larger), the area a rapidly and asymptotically approaches . If θ ≪ 1 {\displaystyle \theta \ll 1} , a = 2 3 c ⋅ h {\displaystyle a={\tfrac {2}{3}}c\cdot h} is a substantially good approximation.
The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk) is based on a similar method. Given a circle of radius r, it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to ...