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The most efficient way to pack different-sized circles together is not obvious. In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an ...
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid 's Elements.
A circular sector is shaded in green. Its curved boundary of length L is a circular arc. A circular arc is the arc of a circle between a pair of distinct points.If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle ...
Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits—a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x 2 + y 2 = 1 ...
For every fraction p / q (in its lowest terms) there is a Ford circle C[p / q], which is the circle with radius 1/(2q 2) and centre at (p / q, 1 / 2q 2 ). Two Ford circles for different fractions are either disjoint or they are tangent to one another—two Ford circles never intersect.
In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
The fraction of points inside the circle approaches π/4 as points are added. Pi can be obtained from a circle if its radius and area are known using the relationship: A = π r 2 . {\displaystyle A=\pi r^{2}.}
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.