Circles

Circle Geometry Notes

Circle Vocabulary

  • Radius: A line segment from the center to any point on the circumference.

  • Diameter (d): A line segment that starts and ends on the circle, passing through the center.

  • Circumference (C): The distance around the circle. Formulas:

  • Area (A): The two-dimensional space taken by a circle.

  • Chord: A line segment with endpoints on the circumference of the circle.

  • Tangent: A line that touches the circle at exactly one point (point of tangency).

  • Secant: A line that touches the circle at two points; can go beyond the circle.

Angles and Arcs

  • Central angle: Vertex is at the circle's center; equals the measure of its intercepted arc.

  • Inscribed angle: Vertex is on the circumference; its measure is half that of its intercepted arc.

Arc Definitions

  • Minor Arc: Measures less than 180°.

  • Major Arc: Measures more than 180°.

  • Intercepted Arc: Arc bound by the sides of an angle.

Arc Measurements

  • Arc length can be measured in degrees or radians, as well as in linear units.

Sector Area

  • Sector: Part of a circle bounded by two radii and the arc between them.

Circle Theorems Overview

  • There are various theorems regarding the properties of circles which include relationships between chords, tangents, and angles.

  • Tangent Radius Theorem: The tangent line at any point on the circle is perpendicular to the radius drawn to that point.

  • Angle Theorem: The angle formed by a tangent and a chord is half the measure of the intercepted arc.

Circle Equation

  • The standard equation for a circle: ( (x - h)^2 + (y - k)^2 = r^2 )

    • Where (h, k) is the center of the circle, and r is the radius.

  • Example for the origin: ( x^2 + y^2 = r^2 )