Ever wondered why a bike wheel stays perfectly round even when you ride over bumps?
Think of a circle as the edge of a perfectly round pizza. Every point on that edge is the same distance from the middle of the pizza.
Circle Basics for ICSE Class 9
Key Terms You Must Know
Radius – the distance from the centre of the circle to any point on its edge. Imagine a string tied at the centre and stretched to the rim; that’s the radius.
Diameter – a straight line that passes through the centre and touches the circle at both ends. It’s simply twice the radius, like the full width of the pizza.
Chord – any line segment whose both ends lie on the circle. If the chord also passes through the centre, it becomes the diameter.
Tangent – a line that touches the circle at exactly one point, never crossing it. Picture a pencil just grazing a ball.
Arc – a part of the circle’s edge, like a slice of the pizza crust.
Sector – the “pizza slice” region bounded by two radii and the arc between them.
Important Relationships
- Diameter = 2 × Radius
- Area of a circle = π × Radius² (π is about 3.14)
- Circumference (the full length around) = 2 × π × Radius
Worked Example 1: Finding a Chord Length
Suppose the radius of a circle is 10 cm and the chord is 6 cm away from the centre. How long is the chord?
Draw a line from the centre to the midpoint of the chord; this line is perpendicular to the chord. You now have a right‑angled triangle:
- Hypotenuse = radius = 10 cm
- One leg = distance from centre to chord = 6 cm
- Other leg = half the chord (let’s call it x)
Using Pythagoras’ theorem (a² + b² = c²):
x² + 6² = 10² → x² = 100 – 36 = 64 → x = 8 cm.
Since x is half the chord, the full chord = 2 × 8 = 16 cm.
Worked Example 2: Area of a Sector
A sector has a central angle of 60° in a circle of radius 5 cm. What is its area?
The whole circle is 360°, so the sector is 60/360 = 1/6 of the circle.
Area of whole circle = π × 5² = 25π cm².
Sector area = (1/6) × 25π ≈ 4.17π ≈ 13.1 cm².
Quick Comparison Table
| Term | Definition | Formula / Relation |
|---|---|---|
| Radius (r) | Distance from centre to any point on the circle | r |
| Diameter (d) | Longest chord passing through centre | d = 2r |
| Chord (c) | Segment joining two points on the circle | c = 2√(r² – d²) where d = distance from centre to chord |
| Tangent (t) | Line touching the circle at exactly one point | t ⟂ radius at point of contact |
| Arc length (L) | Part of the circumference | L = (θ/360) × 2πr where θ is central angle |
Common Mistakes to Avoid
- Mixing up radius and diameter – remember diameter is twice the radius.
- Assuming any line through the centre is a tangent – it’s actually a diameter.
- Using the wrong unit for π – keep π as a symbol or use 3.14 for approximate calculations.
📝 Likely Exam Questions
- Question: In a circle of radius 7 cm, a chord is 5 cm from the centre. Find the length of the chord.
Answer: Using right‑triangle method, half‑chord = √(7² – 5²) = √(49 – 25) = √24 ≈ 4.9 cm. Full chord ≈ 9.8 cm. - Question: Prove that the angle subtended by a diameter at any point on the circle is a right angle.
Answer: By Thales’ theorem, a triangle formed by the diameter and any point on the circle is a right‑angled triangle with the right angle opposite the diameter. - Question: Find the area of a sector with a central angle of 45° in a circle of radius 12 cm.
Answer: Sector area = (45/360) × π × 12² = (1/8) × 144π = 18π ≈ 56.5 cm². - Question: A tangent touches a circle at point P. If the radius OP is 9 cm, what is the distance from O to any point Q on the tangent line?
Answer: The distance OP is perpendicular to the tangent. Any point Q on the tangent forms a right triangle with O, so OQ² = OP² + PQ². Without PQ, the answer is expressed as OQ = √(9² + PQ²). - Question: Explain why all radii of a circle are equal.
Answer: By definition, a radius is the distance from the centre to any point on the circle. Since the centre is fixed, every point on the circle is the same distance away, making all radii equal.