Equation of a circle is based on the definition of the circle: A circle is a set of points which are at equal distance from its center.

So the equation of the circle is derived using distance formula.

Let the center of a circle be at a point (h, k) and its radius be r,

Then distance of center from any point (x, y) lying on the circle is given by
$d^2$ = $\sqrt{(x - h)^2 + (y - k)^2}$

As distance between the center and the point on the center = radius = r

we get
r = $\sqrt{(x - h)^2 + (y - k)^2}$

$r^2$ = $(x - h)^2$ + $(y - k)^2$

Circle Equation

Equation of a circle can be written in the form

$x^2 + y^2 + Ax + By + C = 0$

The center – radius form of the equation of the circle is called as standard form of the equation.

$(x - h)^2 + (y - k)^2 = r^2$
Area of a circle is denoted by the letter A .The area of the circle can be calculated using the equation
A = $\pi r^2$Where r = radius of the circle
$\pi$ = $\frac{22}{7}$ or 3.14
The total length of the circle makes its perimeter. It is also called as circumference of the circle.
Circumference can be calculate if the radius or diameter is known using the equation:
C = 2$\pi$r
C = $\pi$d

The distance from any point lying on the circle from the center is called as radius of a circle. The radius is denoted by letter “r”.

Radius measures half the length of the diameter.

r = $\frac{d}{2}$

If the area of the circle is given then the radius can be found using the equation

r = $\sqrt{\frac{A}{\pi}}$

If the circumference is given then the radius can be found using the formula

R = $\frac{C}{2 \pi}$

From the standard equation of a circle,

Example: $x^2 + y^2 = 16$

Comparing the equation with the standard form

$(x - h)^2 + (y - k)^2 = r^2$

Then the radius for this circle is $\sqrt{16}$ = 4.

If the equation of the circle is given in a general form

$x^2 + y^2 + Ax + By + C = 0$

then the radius can be computed using the formula
r = $\sqrt{\frac{A}{2}^2 + \frac{b}{2}^2 - C}$
Diameter is the longest chord of the circle that passes through the center. It is denoted by letter “d”.
d = 2r
If circumference of the circle is known then diameter can be found out by using the equation

d = $\frac{C}{\pi}$

The total length of the circle makes its perimeter. It is also called as circumference of the circle. Circumference can be calculate if the radius or diameter is known using the equation:

C = 2$\pi$r

C = $\pi$d
Circumference of Circle Equation
An arc is a portion of the circle. The intercepted arc of an angle is determined by the two points of intersection of the angle with the circle.

The arc can be measured in radian as measure of the central angle or length of the arc.

If the arc is intercepted by an angle formed by two radii at the center ,then the measure of the arc equals measure of the central angle.

Arc of a Circle Equation
 
In the above diagram, the green arc is the minor arc (smaller arc) and blue arc is the major arc.

Measure of the minor arc AB = measure of its central angle at point O.

Suppose the $\angle$AOB = 60º, then the measure of the arc AB will also be 60º

As m(arc AB) = m$\angle$AOB = 60º

Since the whole circle measures 360º, we can calculate major arc by using the equation

Major arc = 360º - minor arc
So m(arc ACB) = 360º - 60º = 300º.

The angle at the circumference is called as angle subtended by the arc and it measures half the value of the central angle.

m$\angle$ACB = $\frac{1}{2}$ $\angle$AOB

Length of the arc equation:
S = r θ

Where  r = radius and θ = central angle in radian

S = 2$\pi$r $\times$ $\frac{\theta}{360^o}$...if θ is given in degrees.
A 3 dimensional shape with circular surface is called as a sphere. Volume of a sphere is calculated using the equation:

V = $\frac{4}{3}$ $\pi r^3$

Where V represents volume of the sphere

r represents radius of the sphere

$\pi$ =$\frac{22}{7}$ or 3.14
Half of a circle is generally called as semicircle.

Half Circle Equation

Area of a semicircle equals half the area of the circle. Equation for area of half circle is

A = $\frac{\pi r^2}{2}$

Perimeter of the semicircle is the total length along its edge which includes curve as well as the diameter.

P = $\frac{C}{2}$ + d

Where P: perimeter of the half circle
C: circumference of the whole circle
D: diameter of the circle

So, P = $\frac{2 \pi r}{2}$ + 2r

P = $\pi$ r + 2r

In coordinate geometry the equation of the semicircle can be written as  

y = $\sqrt{r^2 - x^2}$

It will give the semicircle above the x axis as negative values for y are not included.

Half Circle Equations
A line that touches the circle at one and only one point is called as tangent to the circle.

The tangent at any point of a circle is always perpendicular to the radius through the point of contact.

Tangent Circle Equation

Suppose the center of the circle and the point of contact of the tangent are known ,then the equation of the tangent line can be determined.

First find the slope of the radius segment joining the tangent to the center of the circle.
Determine the slope of the tangent line and get the equation using slope point form.

Example:

Tangent Circle Equations

Slope of segment PO = $\frac{y_2 - y_1}{x_2 - x_1}$

  Slope = $\frac{4 - 7}{6 - 2}$
 
        = -$\frac{3}{4}$
        
slope of tangent line = $\frac{-1}{slope}$ of PO

    slope = $\frac{-1}{(-3/4)}$
    
        m = $\frac{4}{3}$
        
Equation of tangent line:

y - $y_1$ = m(x - $x_1$)

y - 7 = $\frac{4}{3}$(x - 2)
The given equation of the circle can be solved to get the center and radius.

Solved Example

Question: Determine the center and the radius of the circle from the given equation
(1) $(x - 4)^2$ + $(y + 3)^2$ = 49
Solution:
 
The standard form of the equation of a circle is

$(x - h)^2$ + $(y - k)^2$ = $r^2$

Write the given equation in similar form.

$(x-4)^2 + (y-(-3))^2 = (7)^2$

So comparing with the standard form we get,
H = 4, k = -3 and r = 7

The center of the circle is (4, -3) and its radius is 7units.

(2) ex: $x^2 + y^2 + 8x - 16y - 20 = 0$

Comparing with general form of the equation we get,

A = 8, B = -16, C = -20

r = $\sqrt{(\frac{8}{2})^2 + (\frac{-16}{2})^2 - (-20)}$

r = $\sqrt{4^2 + (-8)^2 = 20}$

r = $\sqrt{16 + 64 + 20}$

r = $\sqrt{100}$ = 10.

Center is (h, k) where h = $\frac{-A}{2}$ = -4 and

 k = -($\frac{B}{2}$) = -(-8) = 8
 
So center for this circle is (-4, 8) and radius = 10 units.