Circles, Arcs & Sectors

Revise Circles, Arcs & Sectors for International Mathematics 0607 (O Level) — revision notes and instant AI marking.

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Cambridge IGCSE · Extended Maths

Circles, Arcs & Sectors

Big idea: A circle is just every point sitting the exact same distance from a centre — and once you know the radius, everything else (circumference, area, arcs, sectors) is just that radius plugged into a formula, sometimes scaled down by a fraction of the full circle.

Summary — What's in this chapter

A circle is the set of all points equidistant (same distance) from one centre point.
Radius (r) is the distance from centre to edge; diameter (d) is twice the radius, straight across through the centre.
π (pi) ≈ 3.14159… is the fixed ratio between a circle's circumference and its diameter — it's the same number for every circle, ever.
Circumference (perimeter of a circle) = π × d, or equivalently 2 × π × r.
Area of a circle = π × r².
An arc is just a curved "slice" of the circumference — a portion of the outer edge.
A sector is a pizza-slice shaped region bounded by two radii and an arc.
Arc length and sector area are both found by taking the fraction of 360° that the angle θ represents, and applying that same fraction to the full circumference or full area.
Always check whether the exam wants an exact answer (in terms of π) or a rounded decimal — this changes how you use your calculator.

Topic 1: Area & Circumference of Circles

What actually IS a circle?

Picture a dog on a lead, walking around a post while the lead stays perfectly taut. The path the dog traces out is a circle — every single point on that path is exactly one lead-length away from the post. That's the whole definition: a circle is made up of all points on a flat surface that are equidistant (the same distance) from one fixed centre point. The "lead length" is the radius.

Two measurements describe the size of any circle:

Radius (r): centre → edge, the "lead length".
Diameter (d): edge → straight through the centre → other edge. It's exactly double the radius, because it's really just two radii glued end to end through the middle.

Radius vs Diameter . . . . . . r . . ●━━━━━● . ← r = centre to edge . d . (one straight line) . ●━━━━━●━━━━━● ← d = edge to edge through centre . . . (= r + r = 2r)

The circumference is simply the fancy word for a circle's perimeter — the total distance you'd walk if you traced the entire outer edge once.

What is π (pi), really?

Here's a genuinely surprising fact: if you take any circle in the universe — a coin, a wheel, a planet's orbit — and divide its circumference by its diameter, you always get the exact same number: approximately 3.14159... This number is called π (pi), and it never repeats or terminates (it's irrational). It's not something someone invented; it's a built-in property of circles themselves.

Why this matters

Because this ratio is fixed, you never need to "measure" a circle's circumference directly — you can always calculate it exactly from the radius or diameter alone, using π.

"in terms of π" "exact value" 32π cm² more

The two core formulae

Exam Tips

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Also in the full note

Topic 2: Arc Lengths & Sector Areas
What to Memorise
Concepts Checklist
Step-by-step: finding circumference
Step-by-step: finding area
Worked Example: Semicircle
Practice Questions
What are arcs and sectors?

What's inside

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Circles, Arcs & Sectors