1. Understanding the Chord–Central Angle Relationship
The Chord Length and Central Angle Theorem states that:
In a circle, the length of a chord depends on the measure of the central angle that subtends it. A larger central angle produces a longer chord.
This creates a direct connection between chord length, radius, and angle size.
2. Chord Length Formula
If a central angle \(\theta\) (in degrees or radians) subtends a chord in a circle of radius \(r\), then the chord length is:
\(\text{Chord Length} = 2r \sin\left(\dfrac{\theta}{2}\right)\)
3. Why the Formula Works
Consider \(\triangle AOB\), where:
- \(O\) is the centre
- \(A\) and \(B\) lie on the circle
- \(\angle AOB = \theta\)
Because \(OA = OB = r\), the triangle is isosceles. Dropping a perpendicular from \(O\) to the midpoint of \(AB\) splits the triangle into two congruent right triangles.
3.1. Right Triangle Derivation
Half of the chord is:
AM = r \sin\left(\dfrac{\theta}{2}\right)
So the full chord is:
AB = 2r\sin\left(\dfrac{\theta}{2}\right)
4. Comparing Chords Using Central Angles
The size of the central angle determines the length of the chord:
4.1. Larger Angle → Longer Chord
If \(\theta_1 > \theta_2\), then the chord corresponding to \(\theta_1\) is longer.
4.2. Equal Angles → Equal Chords
\theta_1 = \theta_2 \Rightarrow \text{Chord}_1 = \text{Chord}_2
5. Special Cases
This theorem helps explain various well-known geometry facts involving circles.
5.1. Diameter Case
If \(\theta = 180^\circ\), the chord becomes a diameter.
AB = 2r
5.2. Small Angle → Small Chord
If \(\theta\) is very small, the chord becomes short and lies close to the circle's boundary.
6. Example
In a circle with radius \(r = 10\text{ cm}\), find the chord subtended by a central angle of \(60^\circ\).
AB = 2r\sin\left(\dfrac{60^\circ}{2}\right) = 20\sin 30^\circ = 20 \times \dfrac{1}{2} = 10\text{ cm}
7. Real-Life Connections
This theorem appears in:
- Ferris wheel cabin spacing
- Clock designs with equal divisions
- Engineering circular plates and gears
- Road curve planning in circular arcs
Wherever equal angles divide a circle, equal chords are formed.