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Examples of cyclic quadrilaterals. In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral. = + In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle).
A convex quadrilateral is cyclic if and only if opposite angles sum to 180°. Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral. Harmonic quadrilateral: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal. Bicentric quadrilateral: it is both tangential and cyclic.
Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well.
Quadrilateral – 4 sides Cyclic quadrilateral; Kite. Rectangle; Rhomboid; Rhombus; Square (regular quadrilateral) Tangential quadrilateral; Trapezoid. Isosceles trapezoid; Trapezus; Pentagon – 5 sides; Hexagon – 6 sides Lemoine hexagon; Heptagon – 7 sides; Octagon – 8 sides; Nonagon – 9 sides; Decagon – 10 sides; Hendecagon – 11 ...
For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure : not itself an element of a polytope, but a diagram showing how the elements meet.
Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area, 12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral.