Circle Definitions and Properties

Key Definitions

• Centre: The exact middle point of the circle.
• Circumference: The total distance around the outside of the circle.
• Radius ($r$): A line from the centre to the circumference.
• Diameter ($d$): A line through the centre connecting two points on the circumference. $d = 2r$.
• Chord: A straight line connecting any two points on the circumference (not necessarily through the centre).
• Tangent: A line that touches the circle at exactly one point — always perpendicular (90°) to the radius at that point.
• Arc: A part of the circumference.
• Sector: A “pizza slice” — two radii and the arc between them.
• Segment: The region between a chord and the arc it cuts off.

Example 1: Radius and Diameter

(a) A clock face has a radius of 15 cm. What is its diameter?
(b) A hula hoop has a diameter of 90 cm. What is its radius?

(a) $d = 2 \times r = 2 \times 15 = \mathbf{30 \text{ cm}}$.

(b) $r = d \div 2 = 90 \div 2 = \mathbf{45 \text{ cm}}$.

Example 2: The Tangent Property

Line AB is tangent to a circle at point P. O is the centre. What is $\angle OPA$?

A tangent is always perpendicular (at 90°) to the radius at the point of contact.

Answer: $\angle OPA = 90^\circ$.

🤔 Common Misunderstandings

• Sector vs. Segment: A sector is the “pizza slice” shape (two radii + arc). A segment is what’s left when you cut a slice off with a straight chord — the region between a chord and the arc.
• Chord vs. Diameter: A diameter is a special chord that must pass through the centre. All diameters are chords, but not all chords are diameters.

📝 Common Exam Mistakes

• Forgetting the tangent property. The 90° angle between tangent and radius is frequently tested — always state this fact explicitly.
• Using radius when diameter is needed (or vice versa) — double-check which value the formula requires before substituting.
• Misidentifying parts on complex diagrams, especially confusing arc/sector/segment.