Angle at the Centre is Twice the Angle at the Circumference
This activity demonstrates one of the fundamental circle theorems: the angle subtended at the centre of a circle is always twice the angle subtended at the circumference, when both stand on the same arc.
Points A and B define a chord. The red angle ∠AOB is measured at the centre O; the blue angle ∠ACB is measured at point C on the circumference. However you move the points, ∠AOB stays exactly double ∠ACB.
How to Use
Interactive Exploration
Drag any of the three points around the circumference. A and B move the chord, changing both angles together. C slides around the circle without changing the blue angle at all — a striking demonstration that every angle standing on the same arc is the same size.
Move C across the chord onto the shorter arc and the centre angle becomes reflex, still exactly double the (now obtuse) angle at the circumference.
Solving Problems
New positions sets a fresh question: the points move, and one of the two angles is replaced by x while the other keeps its value. Work out the missing angle — double it or halve it — then click ∠AOB or ∠ACB to check your answer.
Either angle can be the unknown, so students have to decide each time whether the rule means multiplying by two or dividing by two.
The reveals track the diagram: drag a point while an answer is showing and the value updates with it.
The Style tab changes the arc colours and the background paper.
Teaching Notes
- Points A and B snap to 2° steps so both angles always come out as whole numbers — the doubling is exact rather than approximate.
- Ask students to drag C right round the major arc first. The blue angle does not move: this is the “angles in the same segment” result hiding inside this theorem.
- Special case: drag A and B to opposite ends of a diameter. The centre angle is 180°, so the angle at the circumference must be 90° — the angle in a semicircle.