Symbolism and religious use The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo. Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is ( 2 r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that ( 2 r − x) x = ( y / 2) 2. Solving for r, we find the required result. The parts of a circle are the radius, diameter, circumference, arc, chord, secant, tangent, sector and segment. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle. If the angle subtended by the chord at the centre is 90 °, then ℓ = r √2, where ℓ is the length of the chord, and r is the radius of the circle.

A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF 2 = AC × AD (tangent–secant theorem).

Circumference of a circle

If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem).

The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass. A superellipse has an equation of the form | x a | n + | y b | n = 1 {\displaystyle \left|{\frac {x}{a}}\right|If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. Note: Secant is not a term you are required to know at GCSE, however it is important to note the difference between a chord and a secant.

Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre. Construction through three noncollinear points Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point"). A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon. The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth. [8] Magic circles are part of some traditions of Western esotericism.Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the centre of the circle to which their arc belongs to. History Circular cave paintings in Santa Barbara County, California Circular piece of silk with Mongol images Circles in an old Arabic astronomical drawing. A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.

The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to 256 / 81 (3.16049...) as an approximate value of π. [3]If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs ( D E ⌢ {\displaystyle {\overset {\frown }{DE}}} and B C ⌢ {\displaystyle {\overset {\frown }{BC}}} ). That is, 2 ∠ C A B = ∠ D O E − ∠ B O C {\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}} , where O is the centre of the circle (secant–secant theorem). Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius. A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon. [18] Every regular polygon and every triangle is a tangential polygon.