What Is Circle Class 10th: Complete Definition and Concepts

Definition and Basic Terms of a Circle

In Class 10 Mathematics, understanding what is a circle is essential. A circle is defined as the set of all points in a plane that are at a fixed distance from a fixed point called the centre.

Key terms related to a circle include:

• Centre (O): The fixed point inside the circle.
• Radius (r): The distance from the centre to any point on the circle.
• Diameter (d): The longest chord passing through the centre, $d = 2r$.
• Chord: A line segment with both endpoints on the circle.
• Circumference: The perimeter or boundary length of the circle.
• Arc: A part of the circumference.
• Sector: The area enclosed by two radii and an arc.

These terms form the foundation for solving problems in the Circles chapter.

Important Formulas Related to Circles

For Class 10 students, memorizing and applying formulas correctly is key to scoring well. Here are the essential formulas:

• Circumference of a circle:

$$ C = 2\pi r $$

• Area of a circle:

$$ A = \pi r^2 $$

• Length of an arc:

$$ \text{Arc length} = \frac{\theta}{360} \times 2\pi r $$

where $\theta$ is the central angle in degrees.

• Area of a sector:

$$ \text{Sector area} = \frac{\theta}{360} \times \pi r^2 $$

Worked example:

If the radius of a circle is 7 cm, find its circumference and area.

• Circumference = $2 \times \pi \times 7 = 14\pi$ cm
• Area = $\pi \times 7^2 = 49\pi$ cm²

These formulas are frequently used in NCERT exercises and CBSE exams.

Properties and Theorems of Circles in Class 10

The Circles chapter in Class 10 NCERT introduces important properties and theorems:

• Equal chords subtend equal angles at the centre.
• The perpendicular from the centre to a chord bisects the chord.
• The angle subtended by a diameter at the circumference is a right angle (90°).
• Tangent to a circle is perpendicular to the radius at the point of contact.

Understanding these theorems helps in solving complex problems involving chords, tangents, and arcs.

Example: If a chord of length 24 cm is 7 cm away from the centre, find the radius of the circle.

Using the perpendicular bisector property:

Let radius = $r$, half chord = 12 cm, distance from centre to chord = 7 cm.

By Pythagoras theorem:

$$ r^2 = 7^2 + 12^2 = 49 + 144 = 193 $$

So, $r = \sqrt{193}$ cm.

Difference Between Radius, Diameter and Chord

Understanding the differences between radius, diameter, and chord is crucial for Class 10 students:

A diameter is always a chord, but a chord is not always a diameter. The radius is half the diameter and always connects the centre to the circle.

Tangent to a Circle: Definition and Properties

A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact.

Key properties:

• The tangent is perpendicular to the radius at the point of contact.
• Only one tangent can be drawn to a circle from a point on the circle.
• From an external point, two tangents can be drawn to the circle, and these tangents are equal in length.

Example: If a radius is 5 cm, the tangent at the endpoint of the radius is perpendicular to it, forming a 90° angle.

This property is often used in problems involving lengths and angles related to tangents.

How to Prepare for Circles Chapter in Class 10 NCERT

To excel in the Circles chapter for Class 10:

• Understand definitions and terms clearly.
• Memorize key formulas for circumference, area, arcs, and sectors.
• Practice all NCERT solved examples and exercises thoroughly.
• Draw neat diagrams to visualize problems better.
• Focus on important theorems and their proofs.
• Attempt previous year CBSE questions on circles.

Regular revision and solving varied problems will build confidence for exams.