The Timpson Lyceum · Geometry (2024)

Angles, Arcs, and the Unit Circle – Lecture 15 (Geometry)

Overview

This lecture builds the measurement of angles on a circle from the ground up. Starting with the idea of an arc and its length (defined as a limit of inscribed polygonal paths, exactly as circumference was), we prove a chain of theorems relating central angles to the arcs they subtend. From there we define the degree as $\tfrac{1}{360}$ of a circle and the radian as the angle whose arc length equals one radius, and we derive the arc-length formulas for each. The lecture closes with the inscribed angle theorem (a central angle is twice the inscribed angle on the same arc), the construction of the unit circle, and the definitions of sine and cosine as coordinates on that circle.

Key Concepts

Arc. Given a central angle (vertex at center $O$) cutting the circle at $A$ and $B$, the arc $AB$ is the set of points on the circle interior to the angle, together with $A$ and $B$. For the exterior arc, a third point $C$ is named: arc $ACB$. For a straight angle the interior is ambiguous, so a labeling point must be supplied.

Arc length (definition). Just as circumference is the limit of perimeters of inscribed polygons, arc length is the limit of lengths of inscribed polygonal paths from $A$ to $B$. Breaking the arc into $n$ chords of total length $L_n$, $\text{arc length of } AB = \lim_{n \to \infty} L_n.$ The notation $\overline{AB}$ here denotes the length of arc $AB$ (a number).

Central-angle / arc-length theorems.

• Congruent angles ⇒ equal arcs (Thm 121). Congruent central angles subtend arcs of equal length. The chord triangles formed at each subdivision are congruent (SAS, using equal radii), so $L_n = L_n'$ at every stage and the limits agree.
• Subdivision freedom (Thm 122). Arc length does not depend on how the arc is partitioned; it only requires the largest piece to shrink to zero (the analysis fact behind Riemann sums).
• Arc addition (Thm 123). For adjacent arcs, the measure of the sum is the sum of the measures: $\overline{AC} = \overline{AB} + \overline{BC}$. Follows from $\lim(L_n + L_n') = \lim L_n + \lim L_n'$.
• Order (Thms 124–126). A bigger central angle subtends a longer arc, and conversely; hence two central angles are congruent if and only if their arcs have equal length.
• Proportionality across circles (Thm 127). Congruent angles on circles of radii $r$ and $r'$ subtend arcs proportional to the radii: $\dfrac{\overline{AB}}{\overline{A'B'}} = \dfrac{r}{r'}$. The chord triangles are similar (isosceles with equal apex angles), so each $L_n/L_n' = r/r'$ and the ratio survives the limit. This is what makes a degree (or radian) consistent on every circle.
• Existence (Thm 128). Every arc length between $0$ and the circumference corresponds to some central angle (a completeness/connectedness fact).

Degree. A degree is the central angle subtending $\tfrac{1}{360}$ of the circumference. Its arc length is $\frac{2\pi r}{360} = \frac{\pi r}{180}.$ The $360$ is a historical convention (Babylonian base-60 arithmetic and a 360-day calendar), not mathematically natural. Subunits: $1^\circ = 60'$ (minutes), $1' = 60''$ (seconds). For an angle of $\theta$ degrees the arc length is $\;L = \dfrac{\theta\,\pi r}{180}$, invertible as $\;\theta = \dfrac{180\,L}{\pi r}$.

Radian. A radian is the central angle whose arc length equals one radius. Since the full circumference is $2\pi r$, a full turn is $2\pi$ radians. This gives the clean formulas $L = \theta r, \qquad \theta = \frac{L}{r},$ with no constant factor — the reason radians are used throughout higher mathematics. Key conversions: $2\pi = 360^\circ$, $\pi = 180^\circ$, $\tfrac{\pi}{2}=90^\circ$, $\tfrac{\pi}{3}=60^\circ$, $\tfrac{\pi}{4}=45^\circ$, $\tfrac{\pi}{6}=30^\circ$.

Inscribed angle theorem. An inscribed angle has its vertex on the circle. A central angle is twice any inscribed angle subtending the same arc. Proved in three cases (one side a diameter; center interior to the inscribed angle; center exterior), the first using the isosceles radius triangles plus the fact that an angle inscribed in a semicircle is a right angle, and the other two by adding/subtracting copies of the first case.

Unit circle. The circle of radius $1$ centered at the origin. Standard angles place known coordinates around it, e.g. $(\tfrac{\sqrt2}{2}, \tfrac{\sqrt2}{2})$ at $45^\circ$, $(\tfrac12, \tfrac{\sqrt3}{2})$ at $60^\circ$, $(\tfrac{\sqrt3}{2}, \tfrac12)$ at $30^\circ$, found from the $45$–$45$–$90$ and $30$–$60$–$90$ triangles, then reflected into the other quadrants.

Sine and cosine. Functions of an angle that read off unit-circle coordinates: $\cos\theta$ is the $x$-coordinate, $\sin\theta$ is the $y$-coordinate of the point at angle $\theta$. The others are built from these: $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$, $\sec\theta = \tfrac{1}{\cos\theta}$, $\csc\theta = \tfrac{1}{\sin\theta}$, $\cot\theta = \tfrac{\cos\theta}{\sin\theta}$.

Worked Example

A circular garden path has radius $r = 7$ m. You walk through a central angle of $135^\circ$. How far do you travel, and what angle corresponds to a walk of $\tfrac{21\pi}{4}$ m?

Using the degree arc-length formula, then inverting it:

The same problem in radians is far shorter. With $\theta = \tfrac{\pi}{3}$ and $r = 7$:

Summary

• Arc length is a limit of inscribed chord lengths — the same construction used for circumference.
• Congruent central angles ⇔ equal arcs; bigger angle ⇔ longer arc; adjacent arcs add ($\overline{AC} = \overline{AB} + \overline{BC}$).
• Across different circles, arcs of congruent angles are proportional to the radii, which makes angle units consistent on every circle.
• A degree is $\tfrac{1}{360}$ of a turn (arc length $\tfrac{\pi r}{180}\theta$); a radian is the angle of arc length one radius (arc length $\theta r$) — radians give the cleaner formula.
• A central angle is twice the inscribed angle on the same arc.
• On the unit circle, $\cos\theta$ is the $x$-coordinate and $\sin\theta$ the $y$-coordinate; all other trig functions are built from these.