Circle Theorems

Revise Circle Theorems for International Mathematics 0607 (O Level) — revision notes and instant AI marking.

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Cambridge (CIE) IGCSE · International Maths Extended

Circle Theorems

The big idea: whenever points sit on a circle and you connect them with lines, the angles that form aren't random — they follow strict, provable rules. Learn to spot the shape (arrowhead, bowtie, kite, cyclic quadrilateral) and the angle rule follows automatically.

Quick Summary

Angle at centre = 2 × angle at circumference — same arc, look for an "arrowhead" shape.
Angle in a semicircle = 90° — a special case of the rule above, where the "centre angle" is a straight line (180°).
A radius bisects a chord at right angles (and vice versa) — creates two congruent right-angled triangles.
A radius and a tangent meet at 90° — always perpendicular at the point of contact.
Tangents from the same external point are equal in length — forms a kite with two right angles.
Opposite angles in a cyclic quadrilateral sum to 180° — all four vertices must be on the circumference.
Angles in the same segment are equal — look for a "bowtie" from the same two points.
Alternate Segment Theorem — the angle between a tangent and a chord equals the angle in the alternate segment.

1. Angle at Centre & Circumference

Picture a circle with centre O, and two points P and Q on the circumference. Draw the two radii OP and OQ — that gives you the "angle at the centre." Now pick any third point on the circumference (on the major arc) and join it to both P and Q — that gives you the "angle at the circumference." As long as both angles are subtended by the same arc PQ, the centre angle is always exactly double the circumference angle.

just imagine an arrowhead: X (circumference) / \ / \ P-----Q \ / \ / 2X O (centre — angle here is double)

Circle Theorem

Angle at the centre = 2 × Angle at the circumference

In plain words: if you stand at the centre and look out at an arc, you see twice the angle that someone standing on the far edge of the circle sees, looking at the same arc.

How to spot it: find two radii going to the circumference, then check if there's a third point connected to those same two endpoints. It often looks like an arrowhead pointing into the circle — but it's still true even when the triangle "parts" overlap, or when the lines form a diamond shape (in which case you compare the reflex angle at the centre with the circumference angle).

Worked example: In a circle with centre O, angle OAB = angle OBA = 60° (base angles of an isosceles triangle, since OA = OB = radius), and the angle at the centre (reflex, made of two parts) is 150°. If the angle at the circumference is (x + 60)°, then using the theorem:

2(x + 60) = 150 → 2x + 120 = 150 → 2x = 30 → x = 15

Watch out Don't confuse this with the cyclic quadrilateral rule (opposite angles = 180°). This theorem is about centre vs. circumference from the same arc — not about a quadrilateral at all.

Practice Question 1.1

A and B are points on a circle with centre O. The angle at the centre, angle AOB, is 96°. C is a point on the major arc. Find angle ACB.

Practice Question 1.2

2. Angle in a Semicircle

diameter

Circle Theorem

The angle in a semicircle is 90°

Worked example:

y + 90 + 40 = 180 → y = 50°

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Also in the full note

3. Chords & Tangents
4. Cyclic Quadrilaterals
5. Angles in the Same Segment
6. The Alternate Segment Theorem
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes

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Circle Theorems

Circle Theorems

Quick Summary

1. Angle at Centre & Circumference

2. Angle in a Semicircle

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