An arc of a circle is the part of the circle between two points on the circle. Arcs are associated with the angle they make at the center.
Arc of a Circle

In the above diagram, A, B C and D are points on the circle with center O. The angle at the center ∠AOB cuts the circle at two distinct points A and B, separating the circles into two arcs, the shorter one marked in yellow is called the minor arc and the longer the major arc.

A central angle separates the circle into two parts, each of which is an arc. The measure of each arc is associated with the measure of the central angle.
If the measure of the central angle is less than 180º, then the shorter arc cut of is called the minor arc and its measure is equal to the measure of the central angle. The longer arc is called the major arc and its measure is equal to 360 - the measure of the central angle. If the measure of the central angle is equal to 180º, then the angle cuts the circle into two congruent arcs each of measure 180º.

Example:
In the adjoining diagram, it is given the
measure of ∠AOB = 120º.
Measure of the minor arc $\widetilde{AB}$ = 120º
Measure of the major arc $\widetilde{ACB}$ = 360 -120
                                                                      = 240º.
Arc of a Circle Definition


When the central angle AOB is a straight angle, then it
cuts the circle into two equal arcs each of which is called
a semi circle.
measure of semicircle $\widetilde{ACB}$ = 180º.
measure of semicircle $\widetilde{ADB}$ = 180º.
 Semi-Circle

The measure of an arc is expressed in two forms, angular and linear. The angular measure is generally known as the measure of the arc and the linear measure is called the length of the arc.
The measure of the arc of is related to the measure of the angle that intercept the arc on the circle as shown above.
We have two formulas to find the length of the arc depending on the units used to express the measure of the central angle.
Arc length = x.$\frac{\pi }{180}$.r, where the measure of the central angle = xº and r the radius of the circle.

Arc length = r.θ
, where θ is the measure of the central angle in radians and r the radius of the circle.

Example:
Find the arc length in each case:
1. Measure of the central angle = 240º and radius = 6 cm
2. Measure of the central angle = $\frac{\pi }{2}$ radians and radius = 7 cm.

1. The central angle is given in degree measure.
Hence arc length = x.$\frac{\pi }{180}$.r where x = 240º and r = 6 cm.
= 240 . $\frac{\pi }{180}$ . 6 = 8π cm ≈ 25.13 cm

2. The central angle is given in radians.
Hence arc length = r.θ where r = 7 cm and θ = $\frac{\pi }{2}$ radians.
= 7 . $\frac{\pi }{2}$ = $\frac{7\pi }{2}$ ≈ 11 cm.
If m ∠ AOB < 180º then the minor arc AB is made up of points A and B and all the points on the circle on the interior of the angle AOB. It is written as $\widetilde{AB}$ with an arc sign on top of the letters AB.

Minor Arc

The minor arc is also the shorter arc formed by ∠AOB and can also be written including a point in between A and B as
$\widetilde{ACB}$. But since the major arc and a semicircle are written using three letters, the minor arc is written using only the end points of the arc.

The measure of the minor arc is same as the measure of the central angle AOB.
if m ∠AOB < 180º, then the major arc AB is made up by the points A and B and the points on the circle not on the interior of angle AOB. A major arc is usually named by three points, the two end points and any other point on the arc. In the diagram below major arc AB ($\widetilde{ACB}$) is shown.

Major Arc

The major arc is thus the longer arc formed by ∠ AOB. 
Measure of major arc $\widetilde{ACB}$ = 360 - m ∠AOB.
AC and EB are diameters of circle O. Identify each arc as a major arc, minor arc or a semicircle of the circle. Then find its measure.   

Finding the Arc


1. m$\widetilde{EA}$          2. m$\widetilde{CBD}$     3. m$\widetilde{DC}$   4. m$\widetilde{DEB}$
5. m$\widetilde{AB}$   6. m$\widetilde{CDA}$

1. $\widetilde{EA}$
     The end points of the arc are E and A
     Measure of the central angle = m ∠EOA = m ∠BOC = 57º.       Vertical angles are congruent.
     Since the measure of the central angle is less than 180º, the arc is a minor arc.
     $\widetilde{EA}$  is a minor arc and m$\widetilde{EA}$  = 57º.

2. $\widetilde{CBD}$
     The end points of the arc are C and D.
     Measure of the central angle = m ∠COD = 95º
     The point B is not inside the angle COD. Hence $\widetilde{CBD}$ is a major arc.
     m$\widetilde{CBD}$ = 360 - 95 = 265º.

3. $\widetilde{DC}$
    The end points of the arc are C and D.
     Measure of the central angle = m ∠COD = 95º
     $\widetilde{DC}$ is a minor arc.
     m$\widetilde{DC}$ = 95º.

4. $\widetilde{DEB}$
    The end points of the arc are D and B.
    Measure of the central angle = m ∠DOB = 95 + 57 = 152º.
    The point E is not inside the angle DOB. Hence $\widetilde{DEB}$ is a major arc.
    m$\widetilde{DEB}$ = 360 - 152 = 208º.

5. $\widetilde{AB}$
    The end points of the arc are A and B.
    Measure of the central angle = ∠AOB = ∠EOC = 28 + 95 = 123º.      Vertical angles are congruent.
    $\widetilde{AB}$ is a minor arc.
    m$\widetilde{AB}$ = 123º.

6. $\widetilde{CDA}$
    The end points of the arc are C and A.
     Measure of central angle = m ∠ COA = 180º                  AC is a diameter of the circle.
     $\widetilde{CDA}$ is a semicircle.
     m$\widetilde{CDA}$ = 180º.
Theorem:
If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

Intersecting Circle

Secants AB and CD intersect inside the circle at P. The two angles formed are angles 1 and 2.
Angle 1 and its vertical angle intercept arcs, AC and BD while angle 2 and its vertical angle intercept the arcs, AD and BC.
The two angles are given as half the sum of intercepted arcs as follows:
m ∠1 = `1/2`(m$\widetilde{AC}$ + m$\widetilde{BD}$)
m ∠ 2 = `1/2`(m$\widetilde{AD}$ + m$\widetilde{BC}$)

Similarly arcs are intercepted when two secants or one secant and a tangent or two tangents intersect in the exterior of the circle.
Theorem:
If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

Secant Intersection


Angle formed by the intersection of two secants:

m ∠P = `1/2`(m$\widetilde{RT}$ - m$\widetilde{QS}$)
 Secant Tangent


Angle formed by the intersection of a secant and
a tangent:


m ∠P = `1/2`(m$\widetilde{SQ}$ - m$\widetilde{RQ}$)
 Tangent Circle


Angle formed by the intersection of two tangents:

m ∠P = `1/2`(m$\widetilde{RSQ}$ - $m\widetilde{RQ}$)