Circle Theorems: Angles at the Centre & Circumference

What Are Circle Theorems?

Circle theorems help us understand how angles behave when different lines (like radii, chords, and arcs) are drawn inside a circle.
They’re really useful when solving tricky diagrams that include triangles, arrows, or loops inside a circle.

The Angle at the Centre Is Twice the Angle at the Circumference

This is one of the most important circle theorems.

It tells us:

If two lines (chords) come from the same two points on the edge of a circle:

• One goes to the centre of the circle
• The other goes to another point on the circumference

…then the angle at the centre is twice the angle at the circumference.

To put it plainly:

If angle at the circumference is $x\degree$, then the angle at the centre is:

$$2x\degree$$

What to Look For

To spot this theorem in a question:

• Look for a triangle or arrowhead shape inside a circle
• Check if one angle is at the centre of the circle
• Check if the other angle is on the edge (circumference)
• Make sure both angles are made from the same arc (same two points on the circle)

It still works:

• If the triangle is “pointing backwards”
• If the triangle overlaps itself

• If the angle at the centre is reflex (more than $180\degree$) — it’s still double the angle at the edge

Worked Example

Question:

In the diagram below, angle $\angle BOC = 150\degree$, and angle $\angle OBA = 60\degree$.

Find angle $x = \angle CAO$.

Give reasons for each step.

Step 1:

Look for radii — in this diagram, lines $AO$, $BO$, and $CO$ are radii (equal in length).

That means triangle $ABO$ is isosceles, and so:

$$\angle OAB = \angle OBA = 60\degree \quad \text{(base angles in isosceles triangle)}$$

Step 2:

Now apply the circle theorem:

The angle at the centre is twice the angle at the circumference.

Both angles are formed from the same arc — arc $BC$ — so:

$$\angle BOC = 2 \times (\angle OAB + x)$$ 

$$150 = 2(x + 60)$$

Step 3: Solve

$$150 = 2x + 120$$

$$30 = 2x \quad \Rightarrow \quad x = 15\degree$$

Final Answer:

$$\boxed{x = 15\degree}$$

Try It Yourself!

Have a go at this one:

In a circle with centre $O$, $\angle ABC = 40\degree$.

The angle $\angle AOC$ is at the centre and made from the same arc as $\angle ABC$.

What is $\angle AOC$?