Circles

                                       Circles

Introduction

A circle is a simple closed curve in a plane consisting of all points in the plane that are at a given distance from a given center. The distance between a point on the circle and the center is called the radius. The diameter is the distance across the circle through the center.

Circle with radius and diameterOpens in a new windowwww.calculator.net

Tangent to a Circle

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

Circle with tangent and radiusOpens in a new windowwww.khanacademy.org

Number of Tangents from a Point on a Circle

From a point outside a circle, there are two tangents that can be drawn to the circle. From a point inside the circle, there are no tangents that can be drawn to the circle. From a point on the circle, there is one tangent that can be drawn to the circle.

Circle with two tangents from a point outside the circleOpens in a new windowwww.math-only-math.com

Summary

Circles are important geometric shapes with many applications in the real world. They are used in engineering, architecture, design, and many other fields.

Tangents to circles are also important geometric concepts.

some important formulas related to circles in geometry: Circumference of a Circle (C): C = 2πr, where "r" is the radius of the circle. Area of a Circle (A): A = πr², where "r" is the radius of the circle.

Annulus Area (A): A = π(R² - r²), where "A" is the annulus area, "R" is the radius of the outer circle, and "r" is the radius of the inner circle. Diameter of a Circle (D): D = 2r, where "r" is the radius of the circle. Relation between Circumference and Diameter: C = πD

Segment Area (A): A = (1/2) × r² × (θ - sinθ), where "A" is the segment area, "θ" is the central angle in radians, and "r" is the radius. Arc Length (L): L = (θ/360) × 2πr, where "θ" is the central angle in degrees. Sector Area (A): A = (θ/360) × πr², where "θ" is the central angle in degrees. Chord Length (C): C = 2r × sin(θ/2), where "θ" is the central angle in radians. Segment Area (A): A = (1/2) × r² × (θ - sinθ), where "θ" is the central angle in radians. Tangent to a Circle: The length of the tangent line from an external point to the circle is equal in all directions. Inscribed Angle Theorem: An inscribed angle in a circle is half the measure of the central angle subtended by the same arc. Intersecting Chords Theorem: When two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord. Tangent-Secant Theorem: The product of the length of a secant segment and its external segment is equal to the square of the length of the tangent segment from the same point.

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