Introduction

A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance is called the radius.

Basic Terms Related to Circles

• Radius: The distance from the center of the circle to any point on it.
• Diameter: The longest chord of a circle, equal to twice the radius (D = 2r).
• Chord: A line segment joining any two points on the circle.
• Circumference: The total length of the boundary of the circle, given by 2πr.
• Arc: A part of the circumference.
• Sector: A region enclosed by two radii and an arc.
• Segment: A region enclosed by a chord and the corresponding arc.
• Tangent: A line that touches the circle at exactly one point.
• Secant: A line that intersects the circle at two points.

Properties of a Circle

• Equal chords subtend equal angles at the center.
• The perpendicular from the center to a chord bisects the chord.
• The angle subtended by a semicircle is always 90°.
• The tangents drawn from an external point to a circle are equal in length.

Important Formulas

• Circumference: \(C = 2\pi r\)
• Area: \(A = \pi r^2\)
• Length of an Arc: \(L = (\theta / 360) \times 2\pi r\), where θ is the angle in degrees.
• Area of a Sector: \(A = (\theta / 360) \times \pi r^2\)
• Area of a Segment: \(A = \text{Area of Sector} - \text{Area of Triangle}\)

Tangents to a Circle

A tangent is a line that touches a circle at exactly one point. Key properties include:

• A tangent to a circle is perpendicular to the radius at the point of contact.
• From a point outside the circle, two tangents can be drawn, which are equal in length.

Applications of Circles

• Engineering and construction (e.g., wheels, gears, and bridges).
• Astronomy and planetary motion.
• Art and design (e.g., circular patterns and mandalas).

Conclusion

Circles play a vital role in mathematics and real-world applications. Understanding their properties helps solve various geometrical problems efficiently.