Essential terms for working with circles — the shared vocabulary behind every lesson in this section.
The Vocabulary
The Core Terms, Precisely Defined
This lesson is a reference point rather than a new theorem — it precisely defines terms already used throughout this Circles section, since mixing these up is one of the most common sources of error in circle problems.
Radius: a segment from the center to any point on the circle. Diameter: a chord passing through the center — always exactly twice the radius. Chord: any segment connecting two points on the circle (a diameter is a special chord). Secant: a LINE that intersects the circle at two points, extending infinitely in both directions (a chord is the finite segment between a secant's two intersection points). Tangent: a line touching the circle at exactly ONE point, without crossing into the interior.
Radius (center to edge), chord (edge to edge), secant (crosses through, extends beyond), tangent (touches once)
💡 Memory Trick
"Essential terms for working with circles." Chord vs. secant: a chord is a SEGMENT (finite, with two endpoints on the circle); a secant is a LINE (infinite, but still passing through those same two circle points). Every diameter is a chord, but not every chord is a diameter — only the ones passing through the center.
More Vocabulary
Angles and Regions
Arc: a portion of the circle's curved boundary between two points. Central angle: an angle with its vertex AT the circle's center. Inscribed angle: an angle with its vertex ON the circle, formed by two chords.
Sector: the pie-slice-shaped REGION bounded by two radii and the arc between them (an area). Segment (circular segment — different from a line segment): the region between a chord and the arc it cuts off (also an area, but bounded by a chord instead of two radii).
Using This Vocabulary
Why Precision Here Matters
Every theorem in this Circles section depends on correctly identifying which of these terms applies to a given diagram — the Chord Product Theorem specifically needs two CHORDS (not secants extending outside the circle); Power of a Point specifically distinguishes between secants and tangents; the Inscribed Angle Theorem specifically requires the vertex to be ON the circle, not at the center.
1
Check where the vertex or intersection points are
On the circle, at the center, inside, or outside — this single detail determines which theorem or definition applies.
2
Distinguish segment-based from line-based terms
Chord and radius are finite segments; secant and tangent describe infinite lines (even though a specific tangent SEGMENT can be measured, as in Power of a Point).
3
Distinguish area-based from length-based terms
Sector and (circular) segment are regions with an area; arc is a curved length, not a region.
Full Worked Example
Correctly Identifying Every Labeled Part of a Diagram
Given: A circle diagram shows: a line touching the circle at one point; a segment from the center to the edge; a segment connecting two points on the circle that does NOT pass through the center; and the curved boundary between those same two points. Identify: each part by name.
1
Identify the line touching at one point
A line touching the circle at exactly one point, without crossing in, is a tangent.
2
Identify the center-to-edge segment
A segment from the center to the circle's edge is a radius.
3
Identify the edge-to-edge segment
A segment connecting two points on the circle, NOT through the center, is a chord (specifically not a diameter, since it misses the center).
4
Identify the curved boundary
The curved portion of the circle between those two chord endpoints is an arc.
Correctly naming each part is exactly the skill needed before applying any circle theorem — misidentifying a chord as a secant, for instance, could lead to reaching for the wrong formula.
🎯 Quick Worked Example
A circle has a region bounded by two radii and the arc between them. What is this region called, and is it measured in length or area units?
1
Identify the boundary. Two radii plus an arc — this pie-slice shape is a sector.
2
Determine the measurement type. A sector is a REGION, not just a boundary line.
3
Conclude. A sector is measured in area units (like square units), not length units.
📌 Exam Application
Vocabulary precision is tested constantly through diagram-labeling and multiple-choice questions rather than calculation — the fastest way to avoid losing easy points is to practice quickly distinguishing chord vs. secant, and sector vs. segment, since these particular pairs are the most commonly confused.
⚠️ Most Common Circle Vocabulary Mistakes
Trap 1 — Confusing 'segment' (circular region) with 'segment' (a straight line piece): This vocabulary term is genuinely used two different ways in geometry — a circular segment is a curved REGION near the edge of a circle, while a line segment is simply a finite straight piece of a line; context determines which meaning applies.
Trap 2 — Calling any chord a diameter: A diameter is a SPECIFIC chord — only the one passing through the center. A chord that misses the center is still a chord, but never a diameter.
✓ Quick Self-Test
1) What is the difference between a chord and a secant? 2) What is the difference between a sector and a (circular) segment? 3) Is every diameter a chord? Is every chord a diameter? 4) Where is the vertex of a central angle located? Where is the vertex of an inscribed angle located? 5) Is an arc measured in length or area units?