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The Circle
Definition:
The circle is the geometric locus of those points in the plane that sit at a constant distance (called the radius of the circle) from a fixed point (called the center of the circle).
The circumference is the length of the circle.
The diameter is the distance from one point on the circle to the opposite point directly across the circle's disc, going through the center.
The diameter equals twice the radius.
So the diameter is the length across the circle, while the circumference is the length around the circle.
The area of the circle measures the surface of the whole disc inside the circle.
Circumference, diameter, and the number π
In any circle, the ratio of its circumference to its diameter has always the same numerical value.
The diameter fits approximately three times along the circumference. That is three times with a little bit of room left.
That fixed, constant number, slightly above 3, that shows up in every circle, is called π (pronounced "pie").
This symbol π is the lower case Greek letter equivalent of the English letter "p."
So, the simplest definition of π is:
The number π is an irrational number, and one of the most famous numbers in math since antiquity.
Being an irrational number, π comes with an infinite number of nonrepeating decimals.
π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062.....
There are whole books written about π, as well as memory competitions where people recite very large numbers of π's digits.
There are computer hadware systems specifically built to find more of those digits, plus a myriad internet sites listing thousands of said digits.
In practice though, for most school homework calculations involving π, you will only need to use a few of π's digits.
Either π = 3.1, or π = 3.14, or π = 3.1416 will suffice most of the time, depending on the accuracy required by the problem.
Formulas for the circle
The top two most important circle formulas to know are:
C = 2π r  ( for the circumference ) 
A = π r^{2}  ( for the area ) 
where
r = the radius
It is way better to remember the two above formulas together, rather than each one individually, so you avoid confusing them, or getting their results mixed up.
It is easier to remember them together if you notice both formulas use the same three symbols:
the number 2, the number π, and the variable r for the radius.
The only thing that changes between the two formulas is the placement of the number 2, and its meaning.
In the formula for circumference, the number 2 is located at the front, as a coefficient, indicating a plain multiplication times two.
On the other hand, in the area formula, the number 2 is located at the end, as an exponent for the radius, indicating we must square the radius.
The following table contains twelve formulas applicable to the circle, indicating how to get each one of the four quantities: radius, diameter, circumference, and area, given one of the other three.
Circle Formulas
 From the radius  From the diameter  From the circumference  From the area 
To get the radius 
r = radius 



To get the diameter 

D = Diameter 


To get the circumference 


C = Circumference 

To get the area 



A = Area 
Concentric circles are those with the same center:
Tangent circles are those that touch each other exactly at one point, either both circles being exterior to each other, or one of them being inside the other one:
Orthogonal circles are those that cut each other at 90 degrees, making right angles at two different points on their circumferences:
A circle and a straight line  Tangents, Secants, and Chords
Given a straight line and a circle in the same plane, either the line lies completely outside the circle without touching it, or they intersect each other. When they intersect, there are two possible cases: their intersection consists of either one, or two points.
A tangent to a circle is a straight line that touches it at exactly one point, having the rest of its points outside the circle.
A secant to a circle is a straight line that cuts the circle at two different points.
A chord is the straight line segment of the secant that lies inside the circle.
Chords in a circle  some of their proprties:
 A longest chord in a circle equals the diameter
 Congruent chords are equidistant from the center
 The perpendicular bisector of a chord passes through the center of the circle
 The chord divides the circle into a minor arc and a major arc
 The chord divides the circle's inner disc into a minor segment and a major segment
Circular Arcs, Circular Segments, Circular Sectors
A circular arc is not the whole circle, just a part of it, an open circular line with two endpoints.
A circular segment is the part of area inside the circle limited by a chord, and by one of the circular arcs the chord determines on the circle.
A circular sector is the part of area inside the circle limited by a circular arc, and by the two radiuses emanating from the center and ending at the end points of the circular arc.
A circular segment subtending a central angle less than 180 degrees can be joined with the isoceles triangle having its base at the chord and its vertex at the center of the circle to form a circular sector.
Central Angle and Inscribed Angle
A central angle in a circle has its vertex at the center, and its sides are two of the circle's radiuses.
Each central angle is associated with a corresponding circular arc and a circular sector.
It is said that the central angle subtends its corresponding circular arc.
Central angles and their corresponding circular arcs are the basis for the definition of angle measurement.
An inscribed angle in a circle has its vertex on the circle. The endpoints of its sides are also on the circle.
Each inscribed angle determines three circular arcs, two of them meeting at the vertex of the angle, and one opposed to it.
The inscribed angle is associated with this third circular arc that does not include the vertex.
The central angle subtending the same circular arc as a given inscribed angle, is twice as big as the inscribed angle.
For example, in the circle to the right in the figure above, central angle ACB is twice as big as inscribed angle ADB.
As a consequence, all inscribed angles subtending the same circular arc are congruent:
In the figure above, inscribed angles ADB, AEB, AFB, and AGB are all congruent to each other.
Intersecting chords or secants  segment lengths product formula
Let A, B, C, and D be four points on a circle in such a way that chords AB and CD are not parallel.
Let P be either the point where chords AB and CD intersect each other inside the circle, or the point where their extended secants intersect each other outside the circle.
Then PA ∙ PB = PC ∙ PD, that is, the products of the lengths of the segments of the chords or secants, measured from the intersection point P, are equal to each other.
Inscribed and Circumscribed circles
A circle is inscribed in a polygon when all the polygon's sides are tangent to the circle.
A circle is circumscribed to a polygon when all the polygon's vertices are on the circle.
 Every triangle has an inscribed circle, called its Incircle, and whose center is called the Incenter of the triangle.
 Every triangle has a circumscribed circle, called its Circumcircle, and whose center is called the Circumcenter of the triangle.
 Every regular polygon has an inscribed circle, and a circumscribed one.
 Nonregular polygons with four or more sides may or may not have an inscribed circle, or a circumscribed one.
The circle and the hexagon
Given a circle, it is possible to draw a sequence of six congruent circles around it, all tangent to the central one, and each one being also tangent to the previous one, and to the next one in the sequence, as shown in the figure below.
The centers of these six congruent, tangent circles are the vertices of a regular hexagon.
Conic Sections
The circle is the simplest of the four Conic Sections, the plane curves produced by graphing the solution sets of quadratic polynomial equations in two variables in the XYplane. The other three conic sections are the Ellipse, the Parabola, and the Hyperbola.
General Quadratic Polynomial Equation in Two Variables 

Conic Sections
Circle  Ellipse 


Parabola  Hyperbola 


The circle as intersection of two surfaces in 3Dspace
In threedimensional space, the circle often shows up as a possible intersection of two surfaces, for example:
 Two spheres
 Two cones
 Two ellipsoids
 Two paraboloids
 A sphere and a cone
 A sphere and an ellipsoid
 A sphere and a paraboloid
 A cone and an ellipsoid
 A cone and a paraboloid
 An ellipsoid and a paraboloid
 A plane and a sphere
 A plane and a cylinder
 A plane and a cone
 A plane and an ellipsoid
 A plane and a torus
