Circles

• Definitions: circle, radius, diameter, chord, arc, sector, and segment.
• Relationships between chords, arcs, and angles subtended at the centre and at the circumference.
• The angle in a semicircle is a right angle (90°).
• Cyclic quadrilaterals — opposite angles sum to 180°.
• Formulas for circumference and area, with practical applications.

1. Basic terms — A circle is all points equidistant from centre O. Radius (r) = distance from centre to circle. Diameter (d) = 2r, longest chord through centre.

2. Chord & arc — A chord joins two points on the circle. An arc is the portion of the circle between those points. Equal chords subtend equal arcs.

3. Angle at centre vs circumference — Angle subtended by an arc at the centre is double the angle at any point on the remaining part of the circle.

4. Angle in semicircle — Angle in a semicircle is 90°. If AB is a diameter and C is on the circle, ∠ACB = 90°.

5. Cyclic quadrilateral — All four vertices on a circle. Opposite angles sum to 180°: ∠A + ∠C = 180°, ∠B + ∠D = 180°.

6. Key formulas:

   • Circumference: C = 2πr = πd
   • Area: A = πr²

7. Where it shows up — Wheels and gears, circular tracks, pizza slices (sectors), clock faces, and irrigation sprinklers.

A chord AB subtends 80° at the centre O. Find the angle subtended by the same chord at point P on the major arc.

Step 1 — Angle at centre = 80° (given)
Step 2 — Angle at circumference = half the angle at centre
Step 3 — ∠APB = 80° ÷ 2 = 40°
Answer: 40°

Application: A circular garden has radius 7 m. Fencing cost = circumference × rate = 2π(7) × rate ≈ 44 m of fencing (using π ≈ 22/7).

• Confusing radius with diameter (d = 2r, not r = 2d).
• Applying "angle at centre = 2 × angle at circumference" when the point is on the same arc as the chord (minor arc case).
• Forgetting opposite angles of a cyclic quadrilateral are supplementary, not equal.
• Using diameter instead of radius in A = πr².

• If radius = 14 cm, find circumference (use π = 22/7).
• In a cyclic quadrilateral, ∠P = 75°. Find ∠R.
• AB is a diameter. C is on the circle. If ∠BAC = 35°, find ∠ABC.
• Define a sector of a circle.

Open the Practice tab for graded questions on Circles.