This topic summary covers Knowledge Organiser: Properties of Circles within Properties of Circles for GCSE Mathematics. Revise Properties of Circles in Geometry & Measures for GCSE Mathematics with 10 exam-style questions and 4 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 6 of 6 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Properties of Circles
Key Terms
- Radius: Distance from centre to circumference
- Diameter: Distance across the circle through the centre; d = 2r
- Chord: A straight line joining two points on the circumference
- Tangent: A line that touches the circle at exactly one point
- Arc: Part of the circumference
- Sector: Region bounded by two radii and an arc (pizza slice)
- Segment: Region between a chord and an arc
Must-Know Facts
- Diameter = 2 × radius
- A tangent meets a radius at exactly 90°
- The perpendicular from the centre to a chord bisects the chord
- Two tangents from an external point are equal in length
- All radii in a circle are equal
- Circumference = πd = 2πr; Area = πr²
Key Formulas
- Diameter: d = 2r
- Circumference: C = πd = 2πr
- Area: A = πr²
- Chord bisector: right-angled triangle → use Pythagoras
Common Mistakes
- Radius vs diameter: Always check which is given — area uses r², circumference uses d or 2r
- Tangent direction: A tangent meets the radius at exactly 90° — must state this in proofs
- Chord length: Chord is not the same as diameter unless it passes through the centre
- Perpendicular from centre: The perpendicular from the centre bisects the chord — use this to find missing lengths
Practice questions for Properties of Circles
A triangle is drawn inside a circle with one side being the diameter. What is the size of the angle at the circumference opposite the diameter?
Prove that the angle in a semicircle is 90°, using the theorem that the angle at the centre is twice the angle at the circumference.