Grade 11 Euclidean Geometry • Theorem 2

Angle at Centre = 2 × Angle at Circumference

The Intuition (Plain English)

Imagine an elastic band stretched between two points on a circle. If you pull the band to the centre of the circle, the angle it makes is exactly twice as wide as it would be if you pulled it all the way back to the opposite edge of the circle.

The Golden Rule: The two angles must be subtended by the exact same arc (or chord), and they must point in the same direction.

The Formal Proof

This is one of the 6 examinable proofs. Examiners specifically look for your construction line and the use of the exterior angle of a triangle.

📐 [Insert GeoGebra Diagram] Circle O. Arc AB subtends ∠AOB at centre and ∠C at circumference.

Given: Circle with centre O. Arc AB subtends ∠AOB at the centre and ∠ACB at the circumference.

R.T.P: ∠AOB = 2∠ACB

Construction: Join C to O and extend to D.

Statement	Reason
1. OA = OC	radii
2. ∴ Â = Ĉ₁	∠s opp equal sides
3. Ô₁ = Â + Ĉ₁	ext ∠ of Δ= sum of int opp ∠s
4. ∴ Ô₁ = 2Ĉ₁	(since Â = Ĉ₁)
5. Similarly, Ô₂ = 2Ĉ₂	same logic in ΔOBC
∴ Ô₁ + Ô₂ = 2Ĉ₁ + 2Ĉ₂	adding the equations
∴ ∠AOB = 2(Ĉ₁ + Ĉ₂)	factorising
∴ ∠AOB = 2∠ACB	proven

How to apply this in an exam (The "Isosceles Trap")

This theorem almost always appears alongside isosceles triangles because the lines going from the centre to the circumference are all radii (which are equal in length).

Typical Exam Question:
In a circle with centre O, arc PQ subtends ∠PRQ = 40° at the circumference. Calculate the size of ∠OPQ.

Step-by-Step Solution:

First, find the angle at the centre using the theorem.
∠POQ = 2 × ∠PRQ
∠POQ = 80° (∠ at centre = 2 × ∠ at circumference)
Next, look at ΔPOQ. Notice that OP = OQ because they are both radii. This makes ΔPOQ an isosceles triangle!
Therefore, the base angles are equal: ∠OPQ = ∠OQP (∠s opp equal sides)
Since the angles in a triangle sum to 180°:
∠OPQ = (180° - 80°) / 2
∠OPQ = 50° (sum of ∠s in ∆)

⚠️ Common Exam Traps

The "Bowtie" Trap: Don't confuse this theorem with "angles in the same segment". This theorem requires one angle to be at the CENTRE. If both angles are touching the circumference, they are equal, not double.
The Reflex Angle: If the angle at the circumference is obtuse (greater than 90°), the corresponding angle at the centre is the REFLEX angle (greater than 180°), pointing in the same direction. Students often accidentally use the interior angle instead of the reflex angle.