G9: Circle Parts
Identify and apply definitions of circle: centre, radius, chord, diameter, circumference, tangent, arc, sector, segment
📋 Key Concepts
Definition: A circle is a set of points that are all the same distance (radius) from a fixed point (centre).
📝 Parts of a Circle
| Part | Definition | Diagram Symbol |
|---|---|---|
| Centre | The fixed point in the middle of the circle | Point O |
| Radius | Distance from centre to any point on the circle | Line from O to edge |
| Diameter | Distance across the circle through the centre (2 × radius) | Line through O |
| Circumference | The distance around the circle (perimeter) | The circle itself |
| Chord | A line joining two points on the circle | Line inside circle |
| Arc | Part of the circumference | Curved edge |
| Sector | Area bounded by two radii and an arc | Pizza slice shape |
| Segment | Area bounded by a chord and an arc | Smaller part of circle |
| Tangent | A line that touches the circle at exactly one point | Line touching edge |
📝 Radius and Diameter
Key Relationship: Diameter = 2 × Radius
d = 2r or r = d/2
Example 1
A circle has radius 7 cm. Find the diameter.
Solution:
d = 2r = 2 × 7 = 14 cm
Example 2
The diameter of a circle is 22 cm. Find the radius.
Solution:
r = d/2 = 22/2 = 11 cm
📝 Chords and Tangents
Chord: A line segment joining two points on a circle. The longest chord is the diameter.
Tangent: A line that touches a circle at exactly one point. The tangent is always perpendicular to the radius at that point.
Example 3
A tangent touches a circle at point P. If the centre is O, what is the angle between OP and the tangent?
Solution:
The tangent is perpendicular to the radius at the point of contact.
Angle = 90°
📝 Arcs, Sectors and Segments
Arc: A portion of the circumference. Named by its endpoints (minor arc < 180°, major arc > 180°).
Sector: The region bounded by two radii and an arc. Looks like a pizza slice.
Segment: The region bounded by a chord and an arc. Can be minor or major.
Example 4
A circle has radius 10 cm. Identify the parts: (a) a line from O to the circumference, (b) the curved edge, (c) the area inside.
Solution:
(a) Radius
(b) Circumference (or arc if only part of it)
(c) Area (or sector/segment if bounded by specific lines)
📝 Important Circle Properties
Properties to remember:
- A diameter is the longest chord
- The perpendicular from the centre to a chord bisects the chord
- Tangents from an external point are equal in length
- The radius is perpendicular to the tangent at the point of contact
Example 5
A chord of length 16 cm is drawn in a circle of radius 10 cm. How far is the chord from the centre?
Solution:
The perpendicular from centre bisects the chord.
Half chord = 8 cm
Using Pythagoras: distance² + 8² = 10²
distance² + 64 = 100
distance² = 36
distance = 6 cm
❓ Practice Questions
Q1: A circle has diameter 18 cm. Find the radius.
Q2: What is the name for a line that touches a circle at exactly one point?
Q3: What is the difference between a sector and a segment?
Q4: What angle does a tangent make with the radius at the point of contact?
Q5: A chord is 24 cm long in a circle of radius 13 cm. Find the distance from the chord to the centre.
✅ Answers
- 9 cm
- A tangent
- A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc
- 90° (perpendicular)
- Half chord = 12 cm, distance = √(13² - 12²) = √25 = 5 cm
🎯 Exam Tips
- Learn all the vocabulary - it's essential for circle theorems
- Remember: diameter = 2 × radius
- The longest chord is always the diameter
- Tangents are perpendicular to radii at the point of contact
- Draw clear diagrams with all parts labelled
- Check if you need minor or major arc/sector/segment
🧠 Problem-Solving Strategies
Problem-Solving
Key parts: radius (centre to circumference), diameter (through centre, chord = 2r), chord (joins two points on circumference), tangent (touches circle at one point, perpendicular to radius at that point), arc (part of circumference), sector (pie-slice region), segment (area between chord and arc). A tangent and radius meet at 90 degrees.
Multi-Step Problem
A circle has radius 10 cm. A chord AB is 16 cm long. Find the distance from the centre of the circle to the chord.
Solution: Draw the perpendicular from centre O to midpoint M of chord AB. AM = 16/2 = 8 cm. In triangle OMA: OA = 10 (radius), AM = 8. By Pythagoras: OM = sqrt(100 - 64) = sqrt(36) = 6 cm. The distance from centre to chord is 6 cm.
⚠️ Common Errors
Watch Out!
1. Wrong: Saying the radius is half the chord Correct: The radius is half the DIAMETER. A chord is any line joining two points on the circumference and is not directly related to the radius.
2. Wrong: Drawing a tangent that crosses the circle at two points Correct: A tangent touches the circle at exactly ONE point and is perpendicular to the radius at that point. It never enters the circle.
3. Wrong: Confusing a sector with a segment Correct: A sector is bounded by two radii and an arc (pie slice). A segment is bounded by a chord and an arc (the region between the chord and the arc).
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A circle has centre O and radius 13 cm. Point P is 5 cm from the centre. A tangent from P touches the circle at T. (a) Explain why angle OTP = 90 degrees. (b) Calculate the length PT. (c) A second tangent from P touches the circle at S. Find the perimeter of quadrilateral OTPS.
(a) A tangent meets the radius at the point of contact at 90 degrees. OT is a radius and PT is a tangent, so angle OTP = 90 degrees.
(b) By Pythagoras in triangle OTP: PT = sqrt(OP squared - OT squared) = sqrt(25 - 169) = sqrt(25 - 169). Wait: OP = 5, OT = 13. PT = sqrt(13 squared - 5 squared) = sqrt(169 - 25) = sqrt(144) = 12 cm.
(c) Tangents from an external point are equal, so PS = PT = 12 cm. Also angle OSP = 90 degrees. Perimeter of OTPS = OT + TP + PS + SO = 13 + 12 + 12 + 13 = 50 cm.
Mark scheme: A1 tangent-radius property, M1 Pythagoras, A1 PT = 12cm, M1 equal tangents, A1 perimeter = 50cm, A1 full reasoning
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student draws a circle and marks two points A and B on the circumference. They then draw chord AB and the perpendicular bisector of AB.
(a) Where does the perpendicular bisector of any chord always pass through?
(b) A second chord CD is drawn. The perpendicular bisectors of AB and CD meet at point X. What is point X?
(c) Explain why the perpendicular bisector of a chord must pass through the centre of the circle.
Answers: (a) The centre of the circle. (b) X is the centre of the circle (both perpendicular bisectors pass through the centre, so their intersection IS the centre). (c) For any chord, the triangle formed by the two radii to the endpoints is isosceles. The perpendicular bisector of the base of an isosceles triangle passes through the vertex (the centre), so the perpendicular bisector of any chord passes through the centre.
📝 Exam Questions by Topic
🎬 Video Resources
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