Parts of a Circle
This topic covers the essential vocabulary used to describe the parts of a circle. Mastering these terms is fundamental for tackling more complex ESAT geometry problems involving areas, lengths, and angles.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The diameter is the longest possible chord in a circle and is always exactly twice the length of the radius.
- A chord divides a circle into two regions called segments.
- Two radii divide a circle into two regions called sectors.
- A tangent is a line that touches the circle's circumference at only a single point.
- The terms 'major' and 'minor' distinguish between the larger and smaller of two arcs, sectors, or segments created from a division of the circle.
Diagram
Formulae
d = 2r To find the diameter (d) given the radius (r), or to find the radius given the diameter.
Definitions
- Radius (r)
- A straight line segment from the centre of the circle to any point on its circumference.
- Diameter (d)
- A straight line segment passing through the centre of the circle whose endpoints both lie on the circumference.
- Chord
- A straight line segment whose endpoints both lie on the circumference of the circle. The diameter is a special case of a chord.
- Sector
- A region of a circle enclosed by two radii and the arc between them, resembling a slice of pizza.
- Segment
- A region of a circle enclosed by a chord and the arc it cuts off.
- Arc
- Any portion of the circumference of a circle.
Worked example
Two radii, OA and OB, form an angle of 100 degrees at the centre O of a circle. What are the geometric names for the region bounded by OA, OB, and the smaller part of the circumference between A and B, and for the straight line connecting points A and B?
- 1
Identify the components forming the first shape:
two radii (OA, OB) and an arc (the smaller one between A and B).
- 2
The definition of a region bounded by two radii and an arc is a sector.
Since the angle is 100 degrees (less than 180), it is the smaller of the two possible sectors.
- 3
This shape is therefore a minor sector.
- 4
Identify the second shape:
a straight line connecting two points (A and B) on the circumference.
- 5
The definition of a straight line connecting two points on the circumference is a chord.
- 6
So, the region is a minor sector and the line is a chord.
Answer: The region is a minor sector, and the line is a chord.
Common mistakes
- ×Confusing a sector with a segment. Remember: a 'sector' is bounded by two 'radii' (like a pizza slice), while a 'segment' is bounded by a 'chord'.
- ×Forgetting that a diameter is a special type of chord. Any line through the centre with ends on the circumference is a diameter.
- ×Mistaking a chord for a tangent. A chord cuts through the circle at two points, whereas a tangent only touches the outside at one point.
No-calculator tips
- ✓Use visualization. Think of a 'sector' as a 'slice' of a pie. Think of a 'segment' as the piece you'd get if you made a single straight cut across the pie.
- ✓For major vs. minor, remember that 'major' means the larger part (greater than a semi-circle) and 'minor' means the smaller part.
- ✓When a problem describes parts of a circle, draw a quick, simple sketch to make sure you have correctly identified all the components.