Mathematics Notes for Class 9 Circles

Circles

In this chapter, you will learn about circles, their properties, important terms, and how to solve problems related to circles. By the end of this chapter, you will be able to identify parts of a circle, use formulas, and apply your knowledge to geometry questions.

Key Concepts

Circle: A set of all points in a plane that are at a fixed distance from a fixed point.
Centre: The fixed point from which all points on the circle are equidistant.
Radius: The distance from the centre to any point on the circle.
Diameter: A chord passing through the centre; it is twice the radius.
Chord: A line segment joining any two points on the circle.
Arc: A part of the circumference of the circle.
Circumference: The distance around the circle.
Sector: The region between two radii and the arc.
Segment: The region between a chord and the arc.

Important Properties

All radii of a circle are equal.
Diameter is the longest chord of a circle.
Equal chords are equidistant from the centre.
The perpendicular from the centre to a chord bisects the chord.

Formulas

Circumference: C = 2πr, where r is the radius.
Area: A = πr²

Examples

Example 1: If the radius of a circle is 7 cm, find its circumference.
C = 2 × π × 7 = 44 cm (using π ≈ 22/7)
Example 2: If the radius of a circle is 5 cm, find its area.
A = π × 5 × 5 = 78.5 cm² (using π ≈ 3.14)

Practice Questions

Define radius, diameter, and chord.
Find the circumference of a circle with radius 10 cm.
Find the area of a circle with diameter 12 cm.
If a chord is 8 cm away from the centre of a circle of radius 10 cm, find the length of the chord.
What is the difference between a sector and a segment?

Challenge Yourself

Draw a circle and mark its centre, radius, diameter, and a chord.
Prove that the perpendicular from the centre to a chord bisects the chord.
Find the area of a sector with radius 6 cm and angle 60°.

Did You Know?

The value of π (pi) is approximately 3.1416 and is the same for all circles.
The wheel is one of the oldest inventions based on the circle.

Glossary

Circle: A round shape with all points equidistant from the centre.
Radius: Distance from the centre to the edge of the circle.
Diameter: A chord passing through the centre; twice the radius.
Chord: A line joining two points on the circle.
Arc: A part of the circle's circumference.
Sector: Area between two radii and the arc.
Segment: Area between a chord and the arc.

Answers to Practice Questions

Radius: Distance from centre to any point on the circle. Diameter: Chord passing through the centre, twice the radius. Chord: Line joining any two points on the circle.
C = 2 × π × 10 = 62.8 cm (using π ≈ 3.14)
Radius = 6 cm; Area = π × 6 × 6 = 113.04 cm²
Use Pythagoras theorem: Chord length = 2 × √(r² - d²) = 2 × √(100 - 64) = 2 × 6 = 12 cm
A sector is the region between two radii and the arc; a segment is the region between a chord and the arc.

Circles are everywhere! Practice drawing and solving problems to master this important topic.

Mathematics Class 9 - Circles Notes