The Timpson Lyceum · Geometry (2024)

Circles, Regular Polygons, and Limits – Lecture 14 (Geometry)

Circles, Regular Polygons, and Limits – Lecture 14

Overview

This lecture connects circles and regular polygons, then uses limits to define a circle's circumference and area. We first establish that every triangle and every regular polygon has both a unique circumscribing circle and a unique inscribed circle, sharing a common center. We develop the tangent-line theorems for circles and the radius/apothem of a regular polygon, then introduce sequences and limits as the tool for passing from polygons to circles. Finally, by inscribing regular polygons with ever-more sides, we define and derive the formulas for circumference and area of a circle, arriving at $C = \pi d$ and $A = \pi r^2$ along with a precise definition of $\pi$ .

Key Concepts

Regular polygon — a polygon in which all sides are congruent and all angles are congruent. (A square is the only regular quadrilateral; a non-square parallelogram with congruent sides — a rhombus — is not regular because its angles differ.)

Circumscribed circle — a circle passing through every vertex of a polygon. The polygon is then inscribed in the circle.

Inscribed circle — a circle lying inside the polygon, tangent to each side. The polygon then circumscribes the circle.

Circumscribing circles exist for special figures only. Every triangle and every regular polygon has one, but a general polygon need not have a circle through all its vertices.

Perpendicular bisector characterization — A point is equidistant from $A$ and $B$ if and only if it lies on the perpendicular bisector of $\overline{AB}$ . This is the engine behind locating circle centers.

Existence/uniqueness theorems (circles & polygons):

Every triangle has exactly one circumscribing circle. Its center is where the perpendicular bisectors of two sides meet; that point is equidistant from all three vertices.
Every regular polygon has exactly one circumscribing circle. Proven by an induction-style argument: a circle through three consecutive vertices $A_1, A_2, A_3$ must also pass through $A_4$ , then $A_5$ , and so on around the figure.
Every regular polygon has exactly one inscribed circle, sharing the same center as the circumscribing circle; its points of tangency are the midpoints of the sides.

Tangent-line theorems:

If a line $L$ passes through a point $P$ on a circle with center $O$ and $L \perp \overline{OP}$ , then $L$ is tangent to the circle (meets it at exactly one point). Proved by contradiction: a second intersection point would force a triangle with two right angles, which is impossible.
Converse: if $L$ is tangent to the circle at $P$ , then $L \perp \overline{OP}$ (proved via the contrapositive).

Center, radius, and apothem of a regular polygon:

Center — the common center of the inscribed and circumscribed circles.
Radius $r$ — distance from the center to a vertex (the circumscribing circle's radius).
Apothem $a$ — distance from the center to the midpoint of a side (the inscribed circle's radius).

Similarity of regular polygons — Two regular polygons are similar if and only if they have the same number of sides. With $n$ sides, all corresponding sides are proportional (each polygon's sides are mutually congruent) and all corresponding angles are congruent.

Interior-angle sum — An $n$ -sided polygon's interior angles sum to $(n-2)$ straight angles, since it splits into $n-2$ triangles.

Proportionality theorems (same number of sides):

Perimeters are proportional to radii and to apothems: $\dfrac{P}{P'} = \dfrac{r}{r'} = \dfrac{a}{a'}$ .
Areas are proportional to the squares of the radii, apothems, or sides: $\dfrac{A}{A'} = \dfrac{r^2}{r'^2} = \dfrac{a^2}{a'^2} = \dfrac{s^2}{s'^2}$ .

Sequences and limits:

A sequence is an ordered list of numbers, written compactly as a term formula like $\{2n\} = 2, 4, 6, \dots$ or $\left\{\tfrac{1}{n}\right\} = 1, \tfrac12, \tfrac13, \dots$
$\displaystyle\lim_{n \to \infty} a_n = L$ means the terms get arbitrarily close to $L$ . E.g. $\lim_{n\to\infty}\tfrac{1}{n} = 0$ , while $\{2n\}$ has no limit (it grows without bound) and $\{(-1)^n\}$ has no limit (it alternates). A constant sequence has that constant as its limit.
Limit laws: if $\lim a_n = a$ and $\lim b_n = b$ , then the limit of a sum/product/quotient is the sum/product/quotient of the limits ( $\lim \tfrac{a_n}{b_n} = \tfrac{a}{b}$ requires $b \neq 0$ ).

Area of a regular polygon — $A = \tfrac{1}{2}Pa$ , derived by cutting the $n$ -gon into $n$ congruent triangles each of base $\tfrac{P}{n}$ and height $a$ .

Apothem approaches the radius — As the number of inscribed sides grows, the apothem closes on the radius: $\lim_{n\to\infty} a_n = r$ .

Circumference (definition) — $\displaystyle C = \lim_{n \to \infty} P_n$ , the limit of the perimeters of inscribed regular $n$ -gons. Circumferences of two circles are proportional to their radii.

Definition of $\pi$ — For any circle, the ratio of circumference to diameter is constant, and that constant is $\pi$ : $\pi = \frac{C}{d}.$ This is the definition; $3.14159\ldots$ is merely a numerical approximation. The ancient Greeks approximated it via inscribed-polygon perimeters (e.g. $\tfrac{22}{7}$ ); a modern series approximation is $\tfrac{\pi}{4} = 1 - \tfrac13 + \tfrac15 - \tfrac17 + \cdots$ .

Area of a circle — $A = \pi r^2$ , obtained by taking the limit of inscribed-polygon areas.

Worked Example

Derive the area of a circle of radius $r$ . The area is the limit of the areas $S_n$ of inscribed regular $n$ -gons, each of area $\tfrac12 P_n a_n$ . Apply the product limit law, using $\lim P_n = C$ (circumference) and $\lim a_n = r$ , then substitute $C = \pi d = 2\pi r$ :

\begin{aligned} A &= \lim_{n \to \infty} S_n = \lim_{n \to \infty} \tfrac{1}{2} P_n\, a_n \\ &= \tfrac{1}{2}\left(\lim_{n \to \infty} P_n\right)\left(\lim_{n \to \infty} a_n\right) \\ &= \tfrac{1}{2}\, C \, r \\ &= \tfrac{1}{2}(2\pi r)(r) \\ &= \pi r^2. \end{aligned}

As a check, the radius/apothem worked example: for a $2\text{ in} \times 2\text{ in}$ square the apothem is $a = 1\text{ in}$ (center to side midpoint), and the radius is the hypotenuse of a right triangle with legs $1$ and $1$ :

\begin{aligned} r &= \sqrt{a^2 + \left(\tfrac{s}{2}\right)^2} \\ &= \sqrt{1^2 + 1^2} = \sqrt{2}\ \text{in}. \end{aligned}

Summary

A regular polygon has all sides and all angles congruent; every triangle and every regular polygon has a unique circumscribing circle and a unique inscribed circle, which share the same center.
A line through a point of a circle is tangent there if and only if it is perpendicular to the radius drawn to that point.
For a regular polygon, the radius reaches the vertices, the apothem reaches the side midpoints, and its area is $A = \tfrac{1}{2}Pa$ .
Two regular polygons are similar exactly when they have the same number of sides; their perimeters are proportional to radii/apothems, and their areas to the squares of those lengths.
Limits of inscribed regular $n$ -gons define a circle's circumference $C = \lim P_n$ and area $A = \lim S_n$ ; the apothem satisfies $\lim a_n = r$ .
$\pi$ is defined as the constant ratio $C/d$ , giving the familiar formulas $C = \pi d = 2\pi r$ and $A = \pi r^2$ .