1. What is a Central Angle?
A central angle is an angle whose vertex lies at the centre of the circle and whose arms (sides) extend to the circumference.
If the centre is \(O\) and the arms meet the circle at points \(A\) and \(B\), then \(\angle AOB\) is a central angle.
\(\angle AOB = \theta\)
2. Relationship Between a Central Angle and Its Arc
A central angle always subtends (creates) an arc on the circle. The measure of this arc is equal to the measure of the central angle.
So if \(\angle AOB = 60^\circ\), then the arc \(AB\) also measures 60°.
3. Properties of Central Angles
Central angles help determine arc lengths, chord lengths, and the shape of sectors.
3.1. Equal Central Angles → Equal Arcs
If two central angles have the same measure, the arcs they subtend are also equal.
\(\angle AOB = \angle COD \Rightarrow \overset{\frown}{AB} = \overset{\frown}{CD}\)
3.2. Equal Arcs → Equal Central Angles
If two arcs are of equal length, their central angles must also be equal because arc measure corresponds directly to angle measure.
3.3. Central Angle and Radius
Since both arms of the angle are radii, a central angle creates an isosceles triangle inside the circle:
\(OA = OB = r\)
4. Central Angle Formula for Arc Length
The length of an arc depends directly on the central angle that subtends it.
4.1. Arc Length (Angle in Degrees)
\(\text{Arc Length} = \dfrac{\theta}{360^\circ} \times 2\pi r\)
4.2. Example
In a circle of radius \(r = 7\text{ cm}\), a central angle of \(\theta = 90^\circ\) gives:
L = \dfrac{90}{360} \times 2\pi \times 7 = \dfrac{1}{4} \times 14\pi = \dfrac{14\pi}{4}\text{ cm}\)
5. Central Angle and Chord Length
The chord joining the endpoints of the arc is directly related to the central angle. A bigger central angle gives a longer chord.
5.1. Chord Length Formula
If the central angle is \(\theta\) and radius is \(r\), then the chord length is:
\(\text{Chord Length} = 2r\sin\left(\dfrac{\theta}{2}\right)\)
5.2. Example
For a circle of radius \(10\text{ cm}\) and central angle \(60^\circ\):
\text{Chord} = 2 \times 10 \times \sin(30^\circ) = 20 \times \dfrac{1}{2} = 10\text{ cm}\)
6. Central Angles in Real Life
Central angles appear in:
- Clock faces (angle made by hands)
- Pie charts
- Ferris wheels
- Circular park designs
- Rotating machinery
Any situation where rotation or circular movement is involved uses the idea of central angles.