Circles

What is a Circle?

A circle is a 2D shape composed of all points in a plane that are equidistant from a given point, called the center.

Key Terms

• Center: The central point of the circle, equidistant from all points on the circle.

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

• Diameter: A line segment that passes through the center of the circle and has endpoints on the circle. The diameter is twice the length of the radius.   

• Circumference: The distance around the circle.

• Area: The amount of space enclosed by the circle.

Formulas

• Circumference: C = 2πr or C = πd, where π (pi) is a mathematical constant approximately equal to 3.14159.

• Area: A = πr², where r is the radius.

• Area of Sector = (θ / 360°) × πr², θ: the central angle of the sector in degrees.

• Arc Length: Arc Length = (θ / 360°) × 2πr, where:

  • • • θ is the central angle of the arc in degrees

      • r is the radius of the circle

• Area of a Segment: (1/2) * r² * (θ - sin(θ)), where:

  • • • θ is the central angle of the segment in radians

      • r is the radius of the circle

• Equation of a Circle:

  • Standard Form: (x - h)² + (y - k)² = r², where

    • • (h, k) are the coordinates of the center of the circle

      • r is the radius of the circle

Tangents and Chords

• Tangent: A line that touches the circle at exactly one point.

• Chord: A line segment that connects two points on the circle.

Angles in a Circle

• Central Angle: An angle whose vertex is the center of the circle.

• Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle. The measure of an inscribed angle is half the measure of its intercepted arc.   

Applications of Circles

Circles are found everywhere in our daily lives, from wheels and gears to the orbits of planets. They have numerous applications in various fields, including:

• Geometry: Many geometric shapes and constructions are based on circles.

• Trigonometry: Trigonometric functions are defined using the unit circle.

• Physics: Circles are used to describe circular motion and orbits.

• Engineering: Circles are used in the design of wheels, gears, and other mechanical components.