Circles
Lessons in this chapter
About Circles
Circles builds a complete toolkit of circle vocabulary and theorems, one relationship at a time.
Basics of Circles names the core vocabulary: center, radius, diameter, chord, before Arcs and Central Angles relates a central angle directly to the arc it cuts off, and Inscribed Angles and Polygons covers the different (but related) rule for an angle whose vertex sits ON the circle.
Chords and Diameters and Properties of Tangents cover two more special segments: a chord's relationship to its distance from the center, and a tangent line's perpendicular relationship to the radius at the point of tangency.
Angle Relationships in Circles and Segment Relationships in Circles close out the geometric relationships with angles and segments formed by intersecting chords, secants, and tangents, and Equations of Circles applies the distance formula to write a circle’s equation from its center and radius on the coordinate plane.
All eight lessons in this chapter open as Geometry worksheets in the page, including Arcs and Central Angles, Inscribed Angles and Polygons and Equations of Circles. Each question is scored as you answer it, which is worth a lot here, because the rule for an angle depends on where its vertex sits, at the center, on the circle, or outside it, and that is easy to mix up. The practice is free to try, with no login needed for your first 3.
Frequently asked questions
Basics of Circles names the center, radius, diameter and chord, then Arcs and Central Angles ties a central angle to the arc it cuts off and Inscribed Angles and Polygons gives the different rule for a vertex sitting on the circle. Chords and Diameters and Properties of Tangents cover two more special segments, Angle Relationships in Circles and Segment Relationships in Circles handle intersecting chords, secants and tangents, and Equations of Circles writes a circle on the coordinate plane from its center and radius. The chapter sits in the high school Geometry conceptual category (G).
A central angle (with its vertex at the circle’s center) always has the exact same measure as the arc it cuts off.
An inscribed angle (with its vertex ON the circle) is always exactly half the measure of the arc it cuts off: a different rule from a central angle's direct match.
A chord is a segment connecting two points on a circle, and the perpendicular distance from the center to a chord determines the chord's length: chords equidistant from the center are always equal in length.
A tangent line touches a circle at exactly one point, and it’s always perpendicular to the radius drawn to that exact point of tangency.
The angle equals half the SUM of the two arcs it and its vertical angle intercept.