A circle has seven key parts that you need to know for KS3 maths: the radius, diameter, chord, arc, tangent, sector and segment. Understanding each term and its relationship to the others is the foundation for circle calculations in Year 7, 8 and 9.
What is the radius?
The radius is the distance from the centre of the circle to any point on its circumference (the curved edge). Every circle has infinitely many radii, and they are all equal in length — that is what makes a shape a circle.
The plural of radius is radii.
Key fact: if the radius is r, then the diameter d = 2r, and if you know d, then r = d/2.
What is the diameter?
The diameter is a straight line that passes through the centre of the circle and touches the circumference at both ends. It is exactly twice the radius.
| Term | Symbol | Relationship |
|---|---|---|
| Radius | r | Half the diameter |
| Diameter | d | d = 2r |
| Circumference | C | C = πd = 2πr |
The diameter is the longest chord that can be drawn inside a circle. Any other chord will be shorter than the diameter.
What is a chord?
A chord is a straight line that connects two points on the circumference without needing to pass through the centre. The diameter is a special chord that does pass through the centre.
Chords are important in circle theorems at GCSE. At KS3, the key thing to remember is:
- A chord divides the circle into two regions.
- The larger region is called the major segment and the smaller one the minor segment.
- A chord is always shorter than or equal to the diameter.
What is an arc?
An arc is part of the circumference — a curved piece of the circle's edge between two points. Just as a chord divides the circumference into two arcs:
- The minor arc is the shorter curved piece.
- The major arc is the longer curved piece.
At KS3 you will use the arc length formula at a basic level. At GCSE Higher you calculate arc lengths using: arc length = (angle at centre / 360) × πd.
What are a tangent, a sector and a segment?
These three terms are frequently confused. Here is a clear comparison:
| Term | What it is | Visual hint |
|---|---|---|
| Tangent | A straight line that touches the circle at exactly one point (the point of tangency) and does not cross it | Like a table top just touching a ball |
| Sector | The region enclosed by two radii and the arc between them | Like a slice of pie |
| Segment | The region enclosed by a chord and the arc it cuts off | Like a slice cut without going through the centre |
Tangent key property (needed at GCSE): A tangent is always perpendicular (at 90°) to the radius drawn to the point of tangency.
How do major and minor versions differ?
Both sectors and segments come in major and minor versions, depending on the angle at the centre:
- Minor sector / minor segment: the smaller region (angle at centre less than 180°).
- Major sector / major segment: the larger region (angle at centre greater than 180°).
If the angle is exactly 180° the sector becomes a semicircle and the chord becomes the diameter.
How do all these terms connect to circle calculations?
Knowing the vocabulary unlocks the formulae:
| Calculation | Formula | Parts used |
|---|---|---|
| Circumference | C = πd | Diameter |
| Area of circle | A = πr² | Radius |
| Arc length | (θ/360) × πd | Arc, diameter, central angle |
| Area of sector | (θ/360) × πr² | Sector, radius, central angle |
At KS3 you mainly use the first two. The arc and sector formulae are introduced at GCSE. All four formulae depend on correctly identifying the radius or diameter first.
Frequently asked questions
What is the difference between a sector and a segment?
A sector is bounded by two straight lines (radii) and a curved edge (arc) — it looks like a pie slice. A segment is bounded by one straight line (a chord) and one curved edge (the arc it cuts off) — it looks like a bite taken out of a circle. The sector always includes the centre of the circle; the segment does not (unless it is a semicircle).
Is a diameter the same as a chord?
A diameter is a special case of a chord — the one that passes through the centre. Every diameter is a chord, but not every chord is a diameter. A diameter is always the longest possible chord in a given circle.
What does "perpendicular" mean in the context of a tangent?
It means the tangent meets the radius at exactly 90°. If you draw a radius to the point where the tangent touches the circle, those two lines form a right angle. This property is used constantly in GCSE circle theorem proofs.
How many tangents can be drawn from a point outside a circle?
Exactly two. From any external point, you can draw two lines that each touch the circle at precisely one point. These two tangents are equal in length, measured from the external point to the point of tangency. This is another circle theorem you will meet at GCSE.
For step-by-step KS3 geometry support with Professor Pi — visit aitutors.me.