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O Level · Mathematics 0580

Circles, Arcs & Sectors

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Circles, Arcs & Sectors

Master the geometry of circles by understanding that everything is a fraction of 360°

What You'll Learn

  • The four core formulas: Circumference, area, arc length, and sector area.
  • The key principle: A full circle = 360°. Any arc or sector is just a fraction of that.
  • How to work backwards: Given area or arc, find the angle or radius.
  • Exact values: Leaving answers in terms of π instead of using decimals.
  • Combined shapes: Semicircles, segments, and composite figures.

Part 1: Area & Circumference of Circles

🎯 What is a Circle?

A circle is the set of all points in a 2D plane that are the same distance from a single point (the centre). That fixed distance is called the radius (r).

The diameter (d) is simply twice the radius: d = 2r

The circumference is the perimeter of the circle — the distance around it. The relationship between circumference and diameter is always the same, and that ratio is called π (pi), which equals approximately 3.14159…

Key insight: Pi (π) isn't just a random number. It's the answer to "how many diameters long is the circumference?" The answer is always π, for every circle, no matter how big or small.

📐 Circumference Formula

The circumference (perimeter) of a circle can be calculated in two equivalent ways:

Circumference
C = πd
or equivalently
C = 2πr
where d = diameter, r = radius

How to remember which to use:

  • If you're given the diameter, use C = πd
  • If you're given the radius, use C = 2πr (because 2r = d)
Circumference is linear: It's measured in simple units like cm or m, not cm². It's a length around the outside, not an area.

📐 Area Formula

The area of a circle is calculated using the radius:

Area of Circle
A = πr²
where r = radius (always squared)

Why is radius squared? Area is always a 2D measurement, so it involves multiplication of two lengths. In this case, both are r, so we get r².

Area is squared: Always measured in square units like cm² or m². This is different from circumference, which is just cm or m.
✏️ Worked Example 1: Semicircle

Question: Find the area and perimeter of a semicircle with diameter 16 cm. Give answers in terms of π.

Step 1: Find the radius
Diameter = 16 cm, so radius r = 16 ÷ 2 = 8 cm
Step 2: Find the area of the full circle
Use A = πr²
A = π(8)² = π × 64 = 64π cm²
Step 3: Halve it for the semicircle
A semicircle = ½ × 64π = 32π cm²
Step 4: Find the perimeter of the semicircle
The perimeter consists of:
  • The curved part (half the circumference)
  • The straight diameter
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Also in the full note
  • Part 2: Arc Lengths & Sector Areas
  • 🎯 What is an Arc?
  • 🎯 What is a Sector?
  • 💡 The Core Principle: Fractions of a Circle
  • 📐 Arc Length Formula
  • 📐 Sector Area Formula
  • 🧠 What to Memorise
  • ✅ Concept Checklist
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