Definition
A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.
A circle is a simple closed curve which divides the plane into two regions: an interior 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 the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.
FORMULA
Area of a Circle a = pi * r2
Circumference c=2pi *r
where;
pi=3.1416
r=radius
Terminology
A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.
A circle is a simple closed curve which divides the plane into two regions: an interior 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 the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.
FORMULA
Area of a Circle a = pi * r2
Circumference c=2pi *r
where;
pi=3.1416
r=radius
Terminology
- Arc: any connected part of the circle.
- Center: the point equidistant from the points on the circle.
- Chord: a line segment whose endpoints lie on the circle.
- Circumference: the length of one circuit along the circle.
- Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a segment, which is the largest distance between any two points on the circle.
- Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
- Secant: an extended chord, a straight line cutting the circle at two points.
- Sector: a region bounded by two radii and an arc lying between the radii.
- Segment: a region bounded by a chord and an arc lying between the chord's endpoints.
- Semicircle: a region bounded by a diameter and an arc lying between the diameter's endpoints. It is a special case of a segment.
- Tangent: a straight line that touches the circle at a single point.
History
The compass in this 13th-century manuscript is a symbol of God's act ofCreation. Notice also the circular shape of the halo
The word "circle" derives from the Greek κίρκος (kirkos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring". The origins of the words "circus" and "circuit" are closely related.
Circular piece of silk with Mongol images
Circles in an old Arabic astronomicaldrawing.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus.
Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[citation needed]
Some highlights in the history of the circle are:
Tughrul Tower from inside
Properties of a Circle
The compass in this 13th-century manuscript is a symbol of God's act ofCreation. Notice also the circular shape of the halo
The word "circle" derives from the Greek κίρκος (kirkos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring". The origins of the words "circus" and "circuit" are closely related.
Circular piece of silk with Mongol images
Circles in an old Arabic astronomicaldrawing.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus.
Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[citation needed]
Some highlights in the history of the circle are:
- 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256⁄81(3.16049...) as an approximate value of π.
Tughrul Tower from inside
- 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles.
- In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.
- 1880 CE– Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.
Properties of a Circle
- The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
- The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
- All circles are similar.
- A circle's circumference and radius are proportional.
- The area enclosed and the square of its radius are proportional.
- The constants of proportionality are 2π and π, respectively.
- The circle which is centred at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
- Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.
- Chords are equidistant from the centre of a circle if and only if they are equal in length.
- The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
- A perpendicular line from the centre of a circle bisects the chord.
- The line segment (circular segment) through the centre bisecting a chord is perpendicular to the chord.
- If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
- If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
- If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
- An inscribed angle subtended by a diameter is a right angle
- The diameter is the longest chord of the circle.
- If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.
- If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.
- The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).
- The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.
- A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.
- A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.
- Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.
- If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.
- If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1⁄2arc(AQ).
Inscribed angles
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).
Theorems
Secant-secant theorem
Secant-secant theorem
- The chord theorem states that if two chords, CD and EB, intersect at A, then CA × DA = EA × BA.
- If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, thenDC2 = DG × DE. (Tangent-secant theorem.)
- If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE. (Corollary of the tangent-secant theorem.)
- The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).
- If the angle subtended by the chord at the centre is 90 degrees then l = r √2, where l is the length of the chord and r is the radius of the circle.
- 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 (DE and BC). This is the secant-secant theorem.
Sagitta
The sagitta is the vertical segment.
Circles inscribed in or circumscribed about other figures
In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.
About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.
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.
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.
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.
Circle as limiting case of other figures
The circle can be viewed as a limiting case of each of various other figures:
The sagitta is the vertical segment.
- The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.
- Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines:
Circles inscribed in or circumscribed about other figures
In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.
About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.
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.
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.
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.
Circle as limiting case of other figures
The circle can be viewed as a limiting case of each of various other figures:
- A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.
- A superellipse has an equation of the form for positive a, b, and n. A supercircle has b = a. A circle is the special case of a supercircle in which n = 2.
- A Cassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.
- A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.