Closed geometric figures {circle, geometry} have centers and circumferences. Circle equation is (x - h)^2 + (y - k)^2 = r^2, where r is radius. For given perimeters, out of all plane figures, circles bound greatest area {area, circle}. Circle area = pi * r^2, where r is radius.
Plane figures {annulus}| can have ring shape.
Circles can have parts {arc, circle} greater than semicircles {major arc} or less than semicircles {minor arc}. Arcs subtend central angle. Area subtended by circle arc = 0.5 * r^2 * A, where A is angle in radians, and r is radius. Arc length s equals radius r times angle A in radians: s = r*A. In two different circles, for same angle, arc-length ratio is proportional to radii ratio.
Circles can have angles {central angle} between two radii.
Line segments {chord, circle}| can link two circle points. Diameter is longest chord.
2*pi radians equals 360 degrees {circular measure}. pi radians equals 180 degrees. pi/2 radians equals 90 degrees.
distance around circle {circumference}.
Circles have radius-length reciprocal {curvature, circle}.
All curve points can invert {inverse curve}. Operations {inversion, curve} can find inverse curves.
For polygons, circumscribed-circle radius {long radius} is longer than inscribed-circle radius.
Circle perimeter is 2 * pi * radius {perimeter, circle}.
Circles can have inscribed quadrilaterals, which have two diagonals. Diagonal product equals sum of opposite-side products {Ptolemy's theorem} {Ptolemy theorem}.
Circle regions {quadrant, circle}| can include one-quarter circumference and two radii.
Two intersecting lines intersect a circle to make line segments {rectangular properties}.
Chords {secant, circle}| can extend beyond circles.
Circular arc and radii from endpoints make pie-piece figures {sector, circle}| {segment, circle}. Sectors {major sector, circle} {major segment, circle} can be greater than semicircles. Sectors {minor sector, circle} {minor segment, circle} can be less than semicircles.
Diameter ends define half circle {semicircle}|. Angle from circle point to diameter ends is right angle.
Arcs define {subtend}| central angle. Area subtended by arc is 0.5 * r^2 * A, where r is radius and A is arc length in radians. Arc length s equals radius r times angle A in radians: s = r*A. In two different circles, for same angle, arc-length ratio is proportional to radii ratio.
Solids {zone, sphere} {band}| can result when circle sector rotates around sphere diameter that does not pass through sector. Zone is on sphere surface. Right-circular cone connects sphere center to closer zone base.
Solids {cap, circle}| {cap zone} can result when circle sector rotates around sphere diameter that passes through sector. Right-circular cone connects sphere center to cap base.
Circles can make ellipsoids {spheroid} of revolution.
Two imaginary points {circular point at infinity} on line at infinity are common to two circles.
Several points {concyclic point} can lie on same circle.
Line from point intersects circle at two points, to form secant. Distances from point to both points can multiply {power, point}. Point power is negative if point is inside circle and positive if point is outside circle.
For circles, radius midpoints define smaller-circle {inversion circle} centers that intersect first circle at only one point and include larger-circle center. Radius diameters intersect first circles at points {inverse point}. Distance from first intersection to first-circle center times distance from inverse point to first-circle center equals r^2 {inversion constant}.
Circles {concentric circle} can have same center.
Circles {escribed circle} can touch three consecutive polygon sides, if two polygon sides extend.
Circles {great circle}| on spheres can have same radius as sphere.
Equation (x - a)^2 + (y - b)^2 + c^2 = 0 has radius = i*c {imaginary circle}.
Circle can touch three consecutive polygon sides {incircle} {inscribed circle}. Inscribed circle has center {incenter}.
Two circles {orthogonal circle} can intersect at right angles. Curves {orthogonal trajectory} can intersect all curve-family members at right angles.
Plane intersects sphere to make circle {small circle}. Small circle does not include great circle.
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Date Modified: 2022.0225