Circles

A circle is a fundamental geometric shape defined as the set of all points in a plane that are equidistant from a fixed center point. The distance from the center to any point on the circle is called the radius. If you connect any two points on the circle with a straight line passing through the center, you get a diameter. The diameter is twice the length of the radius.

Key Concepts:

• Center:

  • The fixed point in the middle of the circle from which all points on the circle are equidistant.

• Radius:

  • The distance from the center of the circle to any point on the circle.

• Diameter: A straight line passing through the center of the circle and connecting two points on the circle. The diameter is twice the length of the radius.

• Circumference: The perimeter or boundary of the circle. The formula for the circumference (C) is C=2πr, where r is the radius.

• Pi (π): An irrational number approximately equal to 3.14159. It is used in formulas involving circles, such as the circumference and area.

• Arc: A portion of the circumference of the circle. A minor arc is smaller than a semicircle, while a major arc is larger.

• Sector: The region enclosed by two radii and the corresponding arc.

• Chord: A straight line segment whose endpoints are on the circle.

Formulas and Properties:

• Circumference Formula: C=2πr or C=πd

• Area Formula: A=πr^2

• Relationship between Radius, Diameter, and Circumference: C=πd, where d is the diameter.

• Sector Area: The area of a sector is a fraction of the entire circle's area, depending on the central angle.

• Sector Area=Central Angle 360∘×��2

• Sector Area= 360∘ Central Angle ×πr^2

• Circle Theorems: Various theorems describe relationships within circles, such as the inscribed angle theorem, which states that an inscribed angle is half the measure of the arc it intercepts.