"Measurement of a Circle Measurement of a Circle or Dimension of the Circle (Greek: , ""Kuklou metrēsis"") is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle. Any circle with a circumference ""c"" and a radius ""r"" is equal in area with a right triangle with the"
"Generalised circle A generalized circle, also referred to as a ""cline"" or ""circline"", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together. Inversive plane geometry is formulated on the plane extended by one point at infinity. A straight line is then thought of as one of the circles that passes through the asymptotic point at infinity. The fundamental transformations in inversive geometry, the ""inversions"", have the property that they map generalized circles to generalized circles. Möbius transformations,"
"Circumference In geometry, the circumference (from Latin ""circumferentia"", meaning ""carrying around"") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, ""circumference"" may also refer to the edge itself rather than to the"
"Line (geometry) The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the ""[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its"
"line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any two points define a line uniquely, and a unique line can be defined by any two points on it. This can be done with compass alone quite naturally. There is nothing to prove. This construction can be done directly with a compass provided the centers and radii of the two circles are known. Due to the compass-only construction of the center of a circle (given below), it can always be assumed that any circle is described by its center and"
"Trochoid A trochoid (from the Greek word for wheel, ""trochos"") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word ""trochoid"" was coined by Gilles de Roberval. As a circle of radius ""a"" rolls without slipping along a line ""L,"" the center ""C"" moves parallel to ""L,"" and every other point ""P"" in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let ""CP = b"". Parametric equations of the trochoid for which ""L"" is"
"Chord (geometry) A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just ""secant"", is the infinite line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter. Every diameter is a chord, but not every chord is a diameter. The word ""chord"" is from the Latin ""chorda"" meaning ""bowstring"". Among properties of chords of a circle are the following: The midpoints of a"
"Central line (geometry) In geometry central lines are certain special straight lines associated with a plane triangle and lying in the plane of the triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994. Let ""ABC"" be a plane triangle and let ( ""x"" : ""y"" : ""z"" ) be the trilinear coordinates of an arbitrary point"
"have where again formula_12 is the length of the semi-major axis and formula_13 is the eccentricity formula_14 In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph. Circumference In geometry, the circumference (from Latin ""circumferentia"", meaning ""carrying around"") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length"
"Circle A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior"
"Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of ""the"" diameter rather than ""a"" diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius"
"It ""is"" possible to do them with straightedge alone given one circle and its center. Straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and machining industry to check the flatness of machined mating surfaces. True straightness can in some cases be checked by using a laser line level as an optical straightedge: it can illuminate an accurately straight line on a flat surface such as the edge of"
"Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"" is the derivative of ""f"". A similar definition applies to space curves and curves in ""n""-dimensional Euclidean space. As it"
"and an exterior. In everyday use, the term ""circle"" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc. A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. The word ""circle"" derives from the Greek κίρκος/κύκλος (""kirkos/kuklos""), itself a metathesis"
"Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ""p"" on the curve has been traditionally defined as the circle passing through ""p"" and a pair of additional points on the curve infinitesimally close to ""p"". Its center lies on the inner normal line, and its curvature is the same as that of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named ""circulus osculans"" (Latin for ""kissing circle"") by Leibniz."
"on the point on the circle and rotate the compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ""ratio"" (other than 1) of distances to two fixed foci, ""A"" and ""B"". (The set of points where the distances are equal is the perpendicular bisector of segment ""AB"", a line.) That circle is sometimes said to be drawn ""about"" two points. The proof is in two parts. First, one must prove that, given two foci ""A"" and ""B"" and a ratio of distances, any point ""P"" satisfying"
"length of the edge. The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this can not be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle is related to one of the most important mathematical constants. This constant,"
"center of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis. The word ""diameter"" is derived from Greek διάμετρος (""diametros""), ""diameter of a circle"", from διά (""dia""), ""across, through"" and μέτρον (""metron""), ""measure"". It is often abbreviated DIA, dia, d, or ⌀. The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of"
"Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon that has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic). All regular simple polygons, all isosceles trapezoids, all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. Not every"
"Secant line In geometry, a secant of a curve is a line that intersects the curve in at least two (distinct) points. The word ""secant"" comes from the Latin word ""secare"", meaning ""to cut"". In the case of a circle, a secant will intersect the circle in exactly two points and a chord is the line segment determined by these two points, that is the interval on a secant whose endpoints are these points. A straight line can intersect a circle in two, one or zero points. A line intersecting in two points is called a ""secant line"", in one"
"has the Euler line as its directrix. Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Euler showed in 1765 that in any triangle, the"
"because formula_5 Apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word ""apothem"" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the"
"from the empty set symbol ∅, from an (italic) uppercase phi ""Φ"", and from the Nordic vowel Ø. See also slashed zero. In German, the diameter symbol (German """") is also used as an average symbol (""Durchschnittszeichen""). Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called"
"a stationary circle of radius formula_17, have the following properties: where formula_19 is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with ""cusps"" replaced by either ""radial maxima"" or ""radial minima."" Trochoid A trochoid (from the Greek word for wheel, ""trochos"") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word ""trochoid"" was coined by Gilles de Roberval. As a circle of radius ""a"" rolls"
"a circle is defined as ""k"" = ±1/""r"", where ""r"" is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa. The plus sign in ""k"" = ±1/""r"" applies to a circle that is ""externally"" tangent to the other circles, like the three black circles in the image. For an ""internally"" tangent circle like the big red circle, that ""circumscribes"" the other circles, the minus sign applies. If a straight line is considered a degenerate circle with zero curvature (and thus infinite radius), Descartes' theorem also applies to a line and two circles"
"notes on the study of curvature, see For application to maneuvering vehicles see Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ""p"" on the curve has been traditionally defined as the circle passing through ""p"" and a pair of additional points on the curve infinitesimally close to ""p"". Its center lies on the inner normal line, and its curvature is the same as that of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the"
"More generally, there is a ""k""-dimensional tangent space at each point of a ""k""-dimensional manifold in the ""n""-dimensional Euclidean space. Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"""
"Hypercycle (geometry) In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the ""axis"", ""center"", or ""base line"" of the hypercycle. The lines perpendicular to the ""axis"", which is also perpendicular to the hypercycle are called the ""normals"" of"
"in Lie sphere geometry. Tangent lines to circles In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line ""t"" to a circle ""C"" intersects the circle at a single point"
"is not being treated formally. When geometry was first formalised by Euclid in the ""Elements"", he defined a general line (straight or curved) to be ""breadthless length"" with a straight line being a line ""which lies evenly with the points on itself"". These definitions serve little purpose since they use terms which are not, themselves, defined. In fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms,"
"Centre (geometry) In geometry, a centre (or center) (from Greek ""κέντρον"") of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of isometry groups then a centre is a fixed point of all the isometries which move the object onto itself. The centre of a circle is the point equidistant from the points on the edge. Similarly the centre of a sphere is the point equidistant from the points on the surface,"
"describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance ""v"" from the chord to the curve (usually at the chord midpoint) is called a ""versine"" measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length ""L"" goes to zero, the ratio goes to the instantaneous curvature. This usage is especially common in rail transport, where it describes measurements of the straightness"
"then circumscribe a square. Again circumscribe a circle, then circumscribe a regular 5-gon, and so on. The radii of the circumscribed circles converge to the so-called ""polygon circumscribing constant"" . The reciprocal of this constant is the Kepler–Bouwkamp constant. Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon that has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic)."