Geometry Reference

Circles

Circle geometry uses a small vocabulary of lines and angle relationships. The key distinction is whether the angle is at the center, on the circle, inside the circle, or outside the circle.

Fact Table

Object	Meaning	Useful Fact
Radius	Segment from center to circle.	All radii of the same circle are congruent.
Diameter	Chord through the center.	Twice the radius; subtends a right angle on the circle.
Chord	Segment with endpoints on the circle.	Congruent chords cut congruent arcs.
Secant	Line that cuts a circle twice.	Creates exterior angle and length relationships.
Tangent	Line that touches a circle once.	Perpendicular to the radius at the point of tangency.
Central angle	Vertex at the center.	Equals its intercepted arc.
Inscribed angle	Vertex on the circle.	Half its intercepted arc.

Content Formulas

Central Angle

$$m\angle=\text{intercepted arc}$$

Inscribed Angle

$$m\angle=\frac12(\text{intercepted arc})$$

Interior Chords

$$m\angle=\frac12(\text{arc}_1+\text{arc}_2)$$

Exterior Secants or Tangents

$$m\angle=\frac12(\text{far arc}-\text{near arc})$$

The lines through a circle are usually radius, diameter, chord, secant, or tangent. An angle bisector is a different object: it cuts an angle into two congruent angles.

Classic Examples

Inscribed Angle

An inscribed angle intercepts an arc of $118^\circ$. Find the angle.

Geometry MoveAn inscribed angle is half its intercepted arc.

$$m\angle=\frac12(118^\circ)$$ $$=59^\circ$$

Exterior Angle

Two secants form an exterior angle. The far arc is $160^\circ$ and the near arc is $64^\circ$. Find the angle.

Geometry MoveExterior circle angles use half the difference of intercepted arcs.

$$m\angle=\frac12(160^\circ-64^\circ)$$ $$=\frac12(96^\circ)$$ $$=48^\circ$$

Tangent and Radius

A tangent touches a circle at point $T$, and $OT$ is a radius. Find the angle between the tangent and $OT$.

Geometry MoveA tangent is perpendicular to the radius at the point of tangency.

$$m\angle=90^\circ$$