What is a circle?
A circle is all the points that are exactly the same distance from a centre point. That distance is the radius.
Parts of a circle
| Part | Description |
|---|---|
| Centre | The middle point |
| Radius (r) | From the centre to the edge |
| Diameter (d) | Across the circle through the centre: d = 2r |
| Circumference (C) | The distance around |
| Chord | A segment joining any two points on the circle |
| Arc | Part of the circumference |
| Sector | A "pizza slice" between two radii and an arc |
| Segment | The region between a chord and an arc |
| Tangent | A line that touches the circle at exactly one point |
The diameter is the longest possible chord.
The radius–diameter link
- d = 2r
- r = d ÷ 2
A circle with radius 5 cm has diameter 10 cm.
Pi (π)
For every circle, the circumference divided by the diameter is the same number: pi (π).
π = 3.14159265… (it never ends or repeats)
We use π ≈ 3.14 or the π key on a calculator.
Key formulas
| Measure | Formula |
|---|---|
| Circumference | C = πd = 2πr |
| Area | A = πr² |
Example: r = 4 cm.
- C = 2 × π × 4 ≈ 25.1 cm
- A = π × 4² = 16π ≈ 50.3 cm²
Angles in circles
- A full turn around the centre is 360°.
- A semicircle is half a circle, made by a diameter (180°).
- A tangent is perpendicular to the radius at the point where it touches.
- An angle inscribed in a semicircle is always 90° (Thales’ theorem).
Circles in real life
Wheels, clocks, pizza, CDs, coins, round tables, Ferris wheels and the orbits of satellites (nearly) are all circular.
Worked examples
Example 1: A circle has diameter 18 m. Find the radius.
18 ÷ 2 = 9 m.
Example 2: Find the circumference of a circle with radius 7 cm.
C = 2π(7) = 14π ≈ 44.0 cm.
Example 3: A pizza has diameter 30 cm. Find its area.
r = 15 → A = π(15²) = 225π ≈ 707 cm².
Common mistakes
- Using the diameter in A = πr². Halve it first.
- Squaring after multiplying by π incorrectly: π × 4² is 16π, not (4π)².
- Mixing up chord and diameter. Only chords through the centre are diameters.
Try it yourself
- Radius of a circle with diameter 26 mm?
- Circumference of a circle with diameter 10 cm?
- Area of a circle with radius 3 m?
- What’s the name for a line touching a circle at one point?
Answers: 1) 13 mm 2) about 31.4 cm 3) about 28.3 m² 4) a tangent
Quick recap
- Radius is half the diameter.
- Circumference = πd; area = πr².
- Pi is the constant ratio C/d ≈ 3.14.