Library › Mathematics A 4MA1 › Circle properties

O Level · Mathematics A 4MA1

Circle properties

Revise Circle properties for Mathematics A 4MA1 (O Level) — revision notes, 106 practice questions and instant AI marking.

📖 Revision notes · preview

Pearson Edexcel International GCSE Mathematics A (4MA1)

Circle properties

The big idea: Most circle facts come from one shape: two radii make an isosceles triangle. Find the centre, draw in the radii, and the theorems start to show.

Summary — What This Chapter Covers

Know the parts: centre, radius, chord, diameter, circumference, tangent, arc, sector, segment.
Tangents from an external point are equal; a tangent is perpendicular to the radius at the point of contact.
The perpendicular from the centre to a chord bisects the chord (and the converse).
Higher: intersecting chords — internal and external.
Higher: the angle theorems — centre, semicircle, same segment, cyclic quadrilateral, alternate segment.

1. Parts of a Circle

Foundation & Higher

The vocabulary

Term	Meaning
Radius	a line from the centre to the circumference
Diameter	a chord through the centre; diameter = 2 × radius
Chord	a straight line joining two points on the circumference
Circumference	the perimeter of the circle
Tangent	a straight line touching the circle at exactly one point
Arc	part of the circumference
Sector	the region between two radii and an arc (a "pizza slice")
Segment	the region between a chord and an arc

2. Chord and Tangent Properties

Foundation & Higher

Three facts with right angles

These three facts usually lead to a right-angled triangle, so Pythagoras or trigonometry finishes the question.

Facts Two tangents from a point to a circle are equal in length
A tangent is perpendicular to the radius at the point of contact
The line from the centre perpendicular to a chord bisects the chord (and the line from the centre to the midpoint of a chord is perpendicular to it)

Worked Example

PA is a tangent to a circle, centre O, at A. OA = 5 cm and OP = 13 cm. Find the length of PA.

Practice Question 1

A chord of length 16 cm is in a circle of radius 10 cm. How far is the chord from the centre?

3. Intersecting Chords

Higher tier

Internal and external

When two chords AB and CD meet at P — inside the circle, or outside it when the chords are extended — the products of the two parts are equal. Each length is measured from P.

Intersecting chords PA × PB = PC × PD (P inside or outside the circle)
Tangent case: if PT is a tangent from P, PT² = PA × PB

Worked Example

Chords AB and CD meet at P inside a circle. AP = 4 cm, PB = 6 cm and CP = 3 cm. Find PD.

Practice Question 2

P is outside a circle. A line from P meets the circle at A and B, with PA = 4 cm and AB = 5 cm. Another line from P meets the circle at C and D with PC = 3 cm. (a) Find CD. (b) A tangent from P touches the circle at T. Find PT.

4. Angle Theorems and Cyclic Quadrilaterals

Higher tier

Four theorems

A cyclic quadrilateral has all four vertices on the circumference of a circle. Formal proofs are not required, but you must quote these in words as reasons.

Theorems

Worked Example

A, B, C and D lie on a circle, centre O, in that order. Angle BOD (on the side of A) is 140°. Find angle BAD and angle BCD.

Same arc

5. The Alternate Segment Theorem

Higher tier

Tangent meets chord

Memorise

🔓 Read the full Circle properties note → You're seeing the preview · sign in to read it all

What's inside

📖 Revision notes ◉ 106 practice questions 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

Past papers that test Circle properties

Real Mathematics A 4MA1 papers with questions on this topic — open one and get it marked instantly, free.

4MA1 Jun 2026 · Paper 1H · Variant R Edexcel · mark scheme Open → 4MA1 Jun 2026 · Paper 2H Edexcel · mark scheme Open → 4MA1 Nov 2025 · Paper 2H Edexcel · mark scheme Open → 4MA1 Jun 2025 · Paper 1F Edexcel · mark scheme Open → 4MA1 Jun 2025 · Paper 1H · Variant R Edexcel · mark scheme Open → 4MA1 Jun 2025 · Paper 2H · Variant R Edexcel · mark scheme Open →

All Mathematics A 4MA1 past papers →

Read the full Circle properties notes free

That's the preview — create a free account to read the rest, plus flashcards and 106 practice questions with instant AI marking. No credit card.

Unlock the full notes free →

Circle properties

Circle properties

The vocabulary

Three facts with right angles

Internal and external

Four theorems

Tangent meets chord

Past papers that test Circle properties

More Mathematics A topics