Geometry

Circle theorems

Move the points. Discover seven relationships.

Predict, test and explain

Step 1 of 3

Predict

The centre angle standing on an arc is 120°. What is the circumference angle standing on the same arc?

What stays the same?\text{What stays the same?}
AA
BB
CC
OO

Scroll the diagram sideways to see every label.

Explore the diagram before revealing the relationship. Points stay on valid arcs.

InvestigateMove a point around the same segment. Which angle stays fixed?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

A and B lie on a circle with centre O. The smaller angle AOB is 96 degrees. C lies on the major arc AB. Find angle ACB.

Hint

The angle at the centre is twice the angle at the circumference standing on the same minor arc.

Reveal answer

The angle at C is half the central angle.

∠ACB=96∘2=48∘\angle ACB=\frac{96^\circ}{2}=48^\circ

Build an explanation

Hint 1

Identify the chord, diameter or tangent involved before choosing a theorem.

Hint 2

Check which arc or segment each marked angle stands on.

Hint 3

For the same arc, the angle at the centre is twice the angle at the circumference.

Worked example

This example uses fixed values, separate from the diagram controls.

An angle at the centre is 110°. Find the angle at the circumference standing on the same arc.

  1. Use the centre-and-circumference theorem for the same arc.
  2. Divide the central angle by two: 110°/2.

Answer: 55°.

Connect this idea

Watch out: Angles in the same segment are equal; points on opposite segments need a different relationship.