Anatomy of a Circle — Radius, Arc, Sector, Radian, π, and Every Part in Between

hari@dasarpai.com (Dr. Hari Thapliyaal) — Wed, 19 Aug 2026 00:00:00 +0000

Anatomy of a Circle
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Every named part, formula, and diagram — strictly in two dimensions
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A circle looks simple: all points the same distance from a center. Yet the vocabulary around it is rich — radius, chord, arc, sector, segment, radian, π, and more. Each name marks a different geometric object or measurement, and each has its own formula.

This article is a 2D reference guide. For every concept you get a short definition, the standard formula, and a labeled diagram. We stay in the plane throughout. When you need the six trigonometric functions as line lengths on the unit circle, see Six Trigonometric Functions — One Diagram. When you want to rotate or extrude a circle into a sphere or cylinder, see Conic Sections and Their 3D Shapes.

The hero figure above labels the parts that share one construction; later sections zoom in on measurements and formulas.

Six Trigonometric Functions — One Diagram

hari@dasarpai.com (Dr. Hari Thapliyaal) — Tue, 18 Aug 2026 00:00:00 +0000

Six Trigonometric Functions — One Diagram
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All six trig values as lengths on one unit-circle picture
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Most textbooks introduce \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) on the unit circle, then treat \(\cot\theta\), \(\sec\theta\), and \(\csc\theta\) as afterthoughts — “just take the reciprocal.” That works algebraically, but it hides the geometry.

This article follows one diagram: a full unit circle, one angle \(\theta\), one ray from the origin, and six colored segments that are the six trigonometric values when the radius is \(1\). The construction is standard mathematics — similar-triangle geometry on the unit circle with tangent lines at \(x=1\) and \(y=1\). What is less common is drawing all six segments together in one figure; each piece is classical, the synthesis is for teaching.

The figure uses \(\theta = 30^\circ\) on purpose (§11). At that angle every segment has a different length, \(T\) and \(C\) are different points, and the dashed secant ray \(OT\) is visibly shorter than the solid cosecant ray \(OC\).

Reciprocal pairs (\(\cos/\sec\), \(\sin/\csc\), \(\tan/\cot\)) are not mirror images on the diagram — see §8 and §9.

Trigonometry on dasarpAI

Anatomy of a Circle — Radius, Arc, Sector, Radian, π, and Every Part in Between

Anatomy of a Circle #

Every named part, formula, and diagram — strictly in two dimensions #

Six Trigonometric Functions — One Diagram

Six Trigonometric Functions — One Diagram #

All six trig values as lengths on one unit-circle picture #

Anatomy of a Circle
#

Every named part, formula, and diagram — strictly in two dimensions
#

Six Trigonometric Functions — One Diagram
#

All six trig values as lengths on one unit-circle picture
#