Lesson Basic formulas in a circle

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In this lesson we will be looking at the definition, different equations, area and circumference of the circle.

Circle
A circle is the set of all point in a plane that are at constant distance from a fixed point.
Fixed point is called the "center of the circle" and the constant distance is called the 
" radius of the circle".

{{{drawing( 160, 160, -0.5, 6.5, -0.5, 5.0, green(line( 2, 2,4.5,2)),red(circle( 2, 2,.2 )),circle( 2,2,2.5), locate(1.3,2.2, C), locate(3,1.9,r), locate(1.4,3.2,d),line(2,0,2,4))}}}

In the figure above,
C is the center of the circle.
r is the radius of the circle.
d is the diameter of the circle. It is twice of the radius. {{{d=2*r}}}

Circumference of the 'Circle':

Perimeter(circumference) of the circle is given by the formula.

{{{P=2*pi*r}}} 
Where:
P is the circumference of the circle.
r is the radius of the circle.

Area of the 'Circle'

Area of the circle is given by the formula.

{{{A= pi*r^2}}}
Where:
A is the area of the circle.
r is the radius of the circle.

Equation of the 'Circle'

In "x-y coordinate system" ('<A HREF=http://www.algebra.com/~pavlovd/wiki/Cartesian_coordinate_system>Cartesian coordinates</A>'), The equation of the circle with center
(a,b) and radius 'r' is

{{{(x-a)^2+(y-b)^2=r^2}}}

Solution of this equation gives the set of all points ({{{x}}},{{{y}}}) which are at constant distance 
'r' from a fixed point (a,b).

In "Parametric form", the equation of the same circle is written as:

{{{x=a+r*Sin(t)}}}

{{{y=b+r*Cos(t)}}}

Where:
(a,b) is the center of the circle.
'r' is the radius of the circle.
't' is the parameter which can take any real value between "{{{-infinity}}} to +{{{infinity}}}".

Solution of this equation gives the set of all points ({{{x}}},{{{y}}}) which are at constant distance
'r' from a fixed point (a,b).

In "Polar coordinate system", the equation of the same circle is written as:
The general equation for a circle with a center at ({{{r[o]}}}, {{{rho}}}) and radius "R" is

{{{r^2-2*r*ro*Cos(theta - rho)+(ro)^2=R^2}}}

Solution of this equation gives the set of all points (r,{{{theta}}}) which are at constant distance
'R' from a fixed point ({{{r[o]}}}, {{{rho}}}).

For further reading 
on Circle refer to <A HREF=Circle.wikipedia>Wikipedia</A>
on different Coordinate systems refer to <A HREF=http://www.algebra.com/algebra/homework/Length-and-distance/Different-type-of-coordinate-system.lesson> Lesson</A>