Circles: definitions and properties
Before circle theorems, you need the vocabulary. This topic defines the parts of a circle and key relationships used throughout geometry.
Parts of a circle
- Centre — the point equidistant from all points on the circle.
- Radius (r) — line segment from centre to circumference. Plural: radii.
- Diameter (d = 2r) — line segment through centre with both endpoints on the circle. Longest chord.
- Chord — straight line segment with both endpoints on the circle.
- Circumference — the curve itself, or its length (2πr).
- Arc — a portion of the circumference between two points.
- Sector — a "pizza slice" — region bounded by two radii and the arc between them.
- Segment — region bounded by a chord and the arc between its endpoints.
- Tangent — a straight line that touches the circle at exactly one point.
Key length facts
- Circumference = 2πr = πd.
- Diameter = 2 × radius.
- A chord is shorter than the diameter unless it IS the diameter.
Tangent properties
- A tangent meets the circle at one point only.
- The tangent is perpendicular to the radius at the point of contact.
- Two tangents from the same external point have equal length.
Arcs and sectors
- A major arc is the longer of two arcs sharing endpoints; the minor arc is shorter.
- A minor sector is bounded by the minor arc; major sector by the major arc.
- A semicircle is half a circle: arc = πr, sector = ½πr².
Segments
A major segment vs minor segment — divided by a chord. The minor segment is the smaller region.
✦Worked example
A circle has radius 5 cm. (a) Find circumference. (b) State a chord that's also a diameter.
- (a) C = 2π(5) = 10π cm ≈ 31.4 cm.
- (b) Any chord through the centre, length 10 cm.
⚠Common mistakes
- Confusing arc with chord — arc is curved; chord is straight.
- Confusing sector with segment — sector has 2 radii + arc; segment has a chord + arc.
- Forgetting tangent ⊥ radius at point of contact.
- Treating diameter as a separate concept from chord — it's a special chord through the centre.
- Mixing radius with diameter in formulae — circumference uses 2πr OR πd, not 2πd.
➜Try this— Quick check
What is the relationship between a tangent and a radius drawn to the point of contact? Answer: they are perpendicular (90°).
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