Circles

⚡ Key Takeaways

A circle is the set of all points in a plane equidistant from a fixed center point. Key result: a circle of radius $r$ has circumference $2\pi r$ and area $\pi r^2$ ; its standard equation is $(x-h)^2+(y-k)^2=r^2$ . Applies throughout geometry, trigonometry, physics, and engineering. Studied since antiquity (Euclid, Archimedes).

💡 The Core Insight

Every property of a circle flows from a single constraint: constant distance from a center. Fix that distance and that center, and the circumference, area, equation, symmetry, and all angle relationships follow inevitably. The circle is "maximally symmetric" — it looks identical under any rotation about its center and any reflection through a diameter.

DefinitionCircle and Its Parts

A circle with center $C$ and radius $r>0$ is the set

1

\quad \{\,P \in \mathbb{R}^2 : |PC| = r\,\}.

Derivation hint: This is the direct set-builder translation of "all points at distance $r$ from $C$ ." Key terms:

Center ( $C$ ): the fixed equidistant point.
Radius ( $r$ ): the constant distance from center to any point on the circle; also any segment from $C$ to the circle.
Diameter ( $d$ ): a chord through the center, $d = 2r$ .
Chord: a segment joining two points on the circle.
Secant: a line meeting the circle in two points.
Tangent: a line meeting the circle in exactly one point.
Arc: a connected portion of the circle.
Circumference ( $C$ ): the total length of the circle.

DefinitionStandard and General Equations

For center $(h,k)$ and radius $r$ , the standard form is

2

\quad (x-h)^2 + (y-k)^2 = r^2.

Derivation hint: Apply the distance formula $|PC|=r$ between $(x,y)$ and $(h,k)$ , then square both sides. Expanding gives the general form

3

\quad x^2 + y^2 + Dx + Ey + F = 0,

where $D = -2h$ , $E = -2k$ , $F = h^2 + k^2 - r^2$ . Derivation hint: Expand $[2]$ and collect; matching coefficients defines $D,E,F$ . This represents a real circle only when $D^2 + E^2 - 4F > 0$ , since

4

\quad r = \tfrac{1}{2}\sqrt{D^2 + E^2 - 4F}.

Derivation hint: Complete the square in $[3]$ ; the constant left over equals $r^2$ , requiring it positive.

TheoremCircumference and Area

For a circle of radius $r$ ,

5

\quad C = 2\pi r = \pi d, \qquad A = \pi r^2.

Derivation hint: $\pi$ is defined as $C/d$ ; the area follows from the limit of inscribed polygons (see proof).

TheoremInscribed Angle Theorem

An inscribed angle is half the central angle subtending the same arc:

6

\quad \theta_{\text{inscribed}} = \tfrac{1}{2}\,\theta_{\text{central}}.

Derivation hint: Draw a radius to the inscribed vertex creating isosceles triangles; the exterior angle equals the sum of equal base angles.

ProofArea of a Circle via Inscribed Polygons

We show $A = \pi r^2$ by approximating the circle with regular $n$ -gons. 1. Inscribe a regular $n$ -gon in the circle. Split it into $n$ congruent isosceles triangles, each with apex at the center and apex angle $2\pi/n$ . 2. Each triangle has two sides of length $r$ . Its area is $\tfrac{1}{2} r^2 \sin(2\pi/n)$ . 3. Total polygon area: $A_n = n \cdot \tfrac{1}{2} r^2 \sin(2\pi/n) = \tfrac{1}{2} r^2 \cdot n\sin(2\pi/n)$ . 4. As $n \to \infty$ , let $x = 2\pi/n \to 0$ . Then $n\sin(2\pi/n) = 2\pi \cdot \dfrac{\sin x}{x} \to 2\pi$ , since $\lim_{x\to 0}\tfrac{\sin x}{x}=1$ . 5. Therefore $A = \lim_{n\to\infty} A_n = \tfrac{1}{2} r^2 \cdot 2\pi = \pi r^2.$ $\blacksquare$

∎

ProofInscribed Angle Theorem (key case)

Let $O$ be the center, $A$ and $B$ endpoints of an arc, and $P$ a point on the major arc, with $O$ inside angle $APB$ . Draw $PO$ and extend to a point $Q$ . 1. Triangle $OAP$ is isosceles ( $OA = OP = r$ ), so $\angle OAP = \angle OPA = \alpha$ . Exterior angle: $\angle AOQ = 2\alpha$ . 2. Triangle $OBP$ is isosceles ( $OB = OP = r$ ), so $\angle OBP = \angle OPB = \beta$ . Exterior angle: $\angle BOQ = 2\beta$ . 3. The inscribed angle is $\angle APB = \alpha + \beta$ . 4. The central angle is $\angle AOB = \angle AOQ + \angle BOQ = 2\alpha + 2\beta = 2(\alpha+\beta)$ . 5. Hence $\angle APB = \tfrac{1}{2}\angle AOB$ . $\blacksquare$

∎

mathematician · Archimedes of Syracuse | c. 287–212 BCE

Archimedes gave the first rigorous treatment of the circle. In Measurement of a Circle he proved that a circle's area equals that of a right triangle with legs $r$ and $C$ (i.e. $A = \tfrac{1}{2} r C = \pi r^2$ ), using the method of exhaustion — bracketing the circle between inscribed and circumscribed polygons. By computing 96-sided polygons he bounded $\pi$ as $3\tfrac{10}{71} < \pi < 3\tfrac{1}{7}$ , an accuracy unmatched for centuries.

🧠 Intuition

Picture a taut string pinned at one end while a pencil at the other end sweeps around — every point it draws is exactly the string's length from the pin. That is a compass, and that single rule (constant radius) is the circle. Now imagine "unrolling" the circle: a wheel of radius $r$ rolled one full turn lays down a track of length $2\pi r$ . The factor $2\pi$ — about 6.28 — is just how many radius-lengths fit around the rim. The area $\pi r^2$ can be felt by slicing the disk into thin pie wedges and rearranging them into a near-rectangle of height $r$ and width $\pi r$ (half the circumference).

Figure

A circle showing center, radius, diameter, chord, tangent, and an inscribed angle

ExampleFinding Center and Radius from General Form

Given: $x^2 + y^2 - 6x + 8y - 11 = 0$ . Find the center and radius.

Steps: 1. Group: $(x^2 - 6x) + (y^2 + 8y) = 11$ . 2. Complete squares: add $(6/2)^2 = 9$ and $(8/2)^2 = 16$ to both sides. $(x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16$ . 3. Rewrite: $(x-3)^2 + (y+4)^2 = 36$ . 4. Compare with $[2]$ : center $(3,-4)$ , $r^2 = 36$ .

Answer: $\boxed{\text{Center } (3,-4),\ r = 6}$

ExampleArc Length and Sector Area

Given: A circle of radius $r = 10\text{ cm}$ with a central angle of $72^\circ$ . Find the arc length and the sector area.

Steps: 1. Convert to radians: $\theta = 72^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{2\pi}{5} \approx 1.2566\text{ rad}$ . 2. Arc length $s = r\theta = 10 \times 1.2566 = 12.566\text{ cm}$ . 3. Sector area $A = \tfrac{1}{2} r^2 \theta = \tfrac{1}{2}(100)(1.2566) = 62.83\text{ cm}^2$ . (Check: $72^\circ$ is $1/5$ of the full circle; full area $=\pi(10)^2=314.16$ , and $314.16/5=62.83$ ✓.)

Answer: $\boxed{s \approx 12.57\text{ cm}, \quad A \approx 62.83\text{ cm}^2}$

⚠️ Common mistake

$\pi$ is not exactly $22/7$ or $3.14$ . These are approximations. $\pi$ is irrational, with a non-terminating, non-repeating decimal expansion ( $3.14159265\ldots$ ). Using $22/7$ ( $\approx 3.142857$ ) introduces an error of about $0.04\%$ — fine for rough work, wrong for precision tasks.

A "circle" is the curve, not the filled region. The filled interior is a disk. The equation $[2]$ uses " $=$ " for the boundary circle; the inequality $(x-h)^2+(y-k)^2 \le r^2$ describes the disk.

Counterexample

Not every second-degree equation is a circle. The general second-degree equation $x^2 + y^2 + Dx + Ey + F = 0$ fails to be a circle when $D^2 + E^2 - 4F \le 0$ . For example, $x^2 + y^2 + 2 = 0$ rearranges to $(x)^2 + (y)^2 = -2$ , which has no real solutions (an "imaginary circle"). And $x^2 + y^2 = 0$ collapses to the single point $(0,0)$ — a "degenerate circle" of radius zero, not a true circle. Equally, $x^2 + 2y^2 = 1$ is an ellipse, not a circle, because the $x^2$ and $y^2$ coefficients differ.

connection

The circle is the geometric heart of Trigonometry: the unit circle ( $r=1$ ) defines $\cos\theta$ and $\sin\theta$ as the coordinates of a point at angle $\theta$ . It is one of the Conic Sections — the special case of an ellipse with equal axes, obtained by slicing a cone perpendicular to its axis. The constant $\pi$ links it to Radian Measure, where angle is arc length divided by radius. Generalizing to three dimensions yields the Sphere, and to constant curvature yields ideas in Non-Euclidean Geometry. The circle's defining locus property connects it to Coordinate Geometry and the Distance Formula.

🔬 Where this appears

⏳ History

- c. 1650 BCE — The Rhind Mathematical Papyrus (Egypt) approximates a circle's area, implying

\pi \approx 3.16

.

- c. 300 BCE — Euclid's Elements, Book III, systematizes circle theorems (chords, tangents, inscribed angles).

- c. 250 BCE — Archimedes bounds

\pi

between

3\tfrac{10}{71}

and

3\tfrac{1}{7}

using 96-gons.

- c. 150 CE — Ptolemy uses

\pi \approx 3.1416

and a table of chords in the Almagest.

- c. 480 CE — Zu Chongzhi (China) computes

\pi

to seven digits,

\approx 355/113

.

- 1671–1674 — Gregory and Leibniz find the series

\tfrac{\pi}{4} = 1 - \tfrac{1}{3} + \tfrac{1}{5} - \cdots

.

- 1761 — Johann Lambert proves

\pi

is irrational.

- 1882 — Ferdinand von Lindemann proves

\pi

is transcendental, finally settling that squaring the circle with compass and straightedge is impossible.

open-problem

The circle hides surprising depths. Squaring the circle — constructing, with compass and straightedge, a square of area equal to a given circle — was proven impossible in 1882 precisely because $\pi$ is transcendental; yet this took over two millennia to resolve. Open questions remain about $\pi$ itself: it is not known whether $\pi$ is a normal number (whether every digit sequence appears with the expected frequency in its decimal expansion). The Gauss circle problem — counting exactly how many integer lattice points lie inside a circle of radius $r$ — has a known leading term $\pi r^2$ , but the precise order of the error term remains unproven. On a curved surface, a "circle" (set of points equidistant from a center) no longer satisfies $C = 2\pi r$ , opening the door to differential geometry.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also