Central angle, as the name suggests is an angle made at the center. Central angles are formed by the hour, minute and seconds hand of a traditional clock. Indeed the three hands make three central angle at instant of time, each angle formed by any two hands of the clock.

Central angle is an important geometric idea based on which many other concepts are defined and theorems proved. Let us look how the central angle is defined and how it is used in solving geometric problems.

## Central Angle Definition

A Central angle is an angle formed at the center of a circle by two radii of the circle. Thus the center the circle is the vertex of the angle and the radii the sides of the angle. If θ is the measure of a central angle then 0º ≤ θ ≤ 180º.

In the above diagram the central angle AOB is made by the radii OA and OB of the circle O at the center. When the two radii coincide the measure of the central angle = 0º and if they form opposite rays, then the measure of central angle = 180º.
The sum of measures of central angles of a circle which contain no common point between them = 360º, as they all together describe the circle.

In the above diagram, the angles 1, 2, 3 and 4 share the common vertex at the center and they do not share any other common point.
m ∠1 + m ∠2 + m ∠3 + m ∠4 = 360º.

## Central Angle Formula

The central angle is related to measure of an arc. The formulas for finding the measure of major and minor arcs are as follows:

Measure of minor arc $\widetilde{AB}$ = measure of central angle AOB.
Measure of major arc $\widetilde{ACB}$ = 360 - measure of central angle AOB.
While the measure of the central angle is equal to the measure of the minor arc intercepted by the angle, the measure of its reflex angle is equal to the measure of the corresponding major arc.
Measure of the central angle is used in formulas for finding the arc length and area of the sector or wedge.

Central angles in circle graph:
The central angles in a circle graph divide the circle into wedges to represent data. The measure of the central angle is proportional to the data frequency represented as a percent. The formula used to find the central angles in constructing a circle graph are as follows:
Measure of central angle = Frequency expressed as a percent x 360º.

### Solved Example

Question: The following is hypothetical table showing the number of hours students of Grade 10 spend on Home work during the week ends.
 Percent of students and Time spent on Home Work during week ends Time in Hours Percent Less than one hour 5% 1 - 2 hrs 10% 2 - 3 hrs 30% 3 - 4 hrs 35% 4 - 5 hrs 12% More than 5 hrs 8%

Solution:

Let us calculate the central angles that would represent the percent for each data category using the formula,
Measure of central angle = Frequency expressed as a percent x 360º.
Measure of central angle to represent less than one hour =   5 % x 360 = 0.05 x 360 = 18º.
1 - 2 hrs           = 10 % x 360 = 0.10 x 360 = 36º.
2 - 3 hrs           = 30 % x 360 = 0.30 x 360 = 108º
3 - 4 hrs           = 35 % x 360 = 0.35 x 360 = 126º
4 - 5 hrs           = 12 % x 360 = 0.12 x 360 =  43.2º
More than 5 hrs           =  8 % x 360  = 0.08 x 360 =  28.8º
Total                                                                       = 360º.

Thus while the central angle is equal to the measure of the minor arc, it is proportional to the area of sector so formed.

## Central Angle Theorem

As the measure of an arc is related to the central angle, the central angle theorem and its converse are as follows:

Theorem:
In a circle or in congruent circles, if central angles are congruent, then their intercepted arcs are congruent.

Converse:
In a circle or in congruent circles, central angles are congruent if the their intercepted arcs are congruent.

If ∠AOB ≅ ∠PO'Q, then $\widetilde{AB}$ ≅ $\widetilde{PQ}$ and
If $\widetilde{AB}$ ≅ $\widetilde{PQ}$, then ∠AOB ≅ ∠ PO'Q.
The theorem and converse can be proved using the relationship between the measure of an arc and the measure of its corresponding central angle.
The central angle theorem can be stated as one combining the above two theorem as follows:

Two arcs of the same or congruent circles are congruent if and only if their corresponding central angles are congruent.

## Central Angle of a Regular Polygon

The sides of a regular polygon form the chords of the circle circumscribing the circle if it exist.
A regular polygon ( polygon with all sides congruent) can always be inscribed in a circle.
Thus the chords of a regular polygon subtend central angles at the center of the circumcircle.

Since the sides of a regular polygon are congruent, the arcs intercepted by them and hence the central angles are all congruent. In general, a regular polygon of n sides has n congruent central angles.
Thus, the measure of the central angle of a polygon of n sides = $\frac{360}{n}$.
The measure of central angle of an equilateral triangle = $\frac{360}{3}$ = 120º.
The measure of central angle of a square = $\frac{360}{4}$ = 90º.
The measure of central angle of a regular hexagon = $\frac{360}{6}$ = 60º.

## Finding Central Angle

What is the measure of the central angle of a semicircle?

A semicircle is intercepted by a diameter, which is formed by two opposite radii making a straight angle at the center.
Hence, the measure of the central angle of a semicircle = 180º.

### Solved Examples

Question 1: If a major arc measures 230º, find the measure of the central angle which intercepted the arc.
Solution:

Let the measure of the central angle be xº.
Measure of the major arc = 360 - measure of the central angle.
230 = 360 - x        or
x = 360 - 230 = 130º.

Question 2: Find the measure of the central angle of a regular polygon of 12 sides.
Solution:

Measure of central angle of a regular polygon = $\frac{360}{number\ of\ sides}$

Measure of central angle of a regular polygon of 12 sides = $\frac{360}{12}$ = 30º.