1. Definition: a set of equidistant points
Given a point $O$ in the plane and a positive number $r$, the circle with center $O$ and radius $r$ is the set of all points whose distance from $O$ equals exactly $r$.
That single constraint — "the same distance from one chosen point" — does all the work. Once you fix a center and a distance, every point that satisfies the rule is determined, and the result is the smooth, perfectly round curve everyone recognizes.
In coordinates, if the center sits at $(h, k)$ and the radius is $r$, a point $(x, y)$ is on the circle when
$$ (x - h)^2 + (y - k)^2 = r^2 $$This is just the Pythagorean theorem in disguise: the horizontal gap $(x-h)$ and the vertical gap $(y-k)$ form the legs of a right triangle whose hypotenuse is the radius.
The circle is the curve, not the filled-in region inside it. The interior — the set of points whose distance from the center is less than $r$ — is called the disk. Most casual usage blurs the two; precise geometry keeps them apart.
2. The basic parts
A handful of named pieces show up over and over. Learn them once and the rest of geometry reads more cleanly.
- Center — the fixed point everything is measured from. Usually labelled $O$.
- Radius — a segment from the center to any point on the circle. Also the name of its length, $r$.
- Diameter — a chord that passes through the center. Its length is $d = 2r$ — the longest segment that fits inside the disk.
- Chord — any segment whose endpoints both lie on the circle. The diameter is the special case where the chord runs through $O$.
- Arc — a connected piece of the circle itself (a portion of the curve, not the interior). Any two points on the circle split it into two arcs: a shorter minor arc and a longer major arc.
Two more pieces — the sector (a pie-slice bounded by two radii and an arc) and the tangent (a line touching the circle at one point) — get their own sections below.
3. Circumference and $\pi$
The circumference is the distance around the circle — its perimeter. It depends on the radius in the simplest possible way:
$$ C = 2\pi r = \pi d $$The constant $\pi$ ("pi") is not pulled out of a hat. It is defined as the ratio of any circle's circumference to its diameter:
$$ \pi = \frac{C}{d} $$The remarkable thing is that this ratio is the same number for every circle — small or large, drawn on paper or traced by a planet. That universality is what makes $\pi$ worth a Greek letter of its own.
What kind of number is $\pi$?
$\pi$ is irrational: it cannot be written as a ratio of two whole numbers. Its decimal expansion runs forever without ever repeating. The familiar $3.14$ is just a three-digit truncation; the actual value begins
$$ \pi \approx 3.141592653589793\ldots $$and continues with no pattern. It is also transcendental, meaning it isn't a root of any polynomial with rational coefficients — a stronger condition than irrationality, settled by Ferdinand von Lindemann in 1882.
Around 250 BCE, Archimedes squeezed $\pi$ between the perimeters of regular polygons inscribed in and circumscribed around a circle. Pushing the construction up to a 96-sided polygon, he established $\tfrac{223}{71} < \pi < \tfrac{22}{7}$ — already accurate to about three decimal places. Today the digits of $\pi$ have been computed to more than 100 trillion places, mostly as a stress test for hardware.
4. Area
The area of the disk bounded by a circle of radius $r$ is
$$ A = \pi r^2 $$That second power of $r$ matters: doubling the radius quadruples the area, not just doubles it. It's the same scaling that turns a $1\times 1$ square into a $2 \times 2$ square — four times the area, not two.
Where does $\pi r^2$ come from?
There is a beautiful, hand-wavy argument that gives the formula directly. Slice the disk into many thin pie-shaped wedges, like cutting a pizza. Now rearrange the wedges alternately tip-up and tip-down. As the slices get thinner, the rearranged shape gets closer and closer to a rectangle:
- The height of the rectangle is the radius $r$ (each wedge contributes its slanted side, which is just $r$).
- The width is half the circumference, $\tfrac{1}{2}(2\pi r) = \pi r$ (the arcs of the wedges, half on the top edge and half on the bottom).
So the area equals base times height:
$$ A = (\pi r)(r) = \pi r^2 $$This isn't a proper proof — making "as the slices get thinner" precise is a job for calculus — but it captures the essential geometry honestly.
The same $\pi$ appears in both formulas, $C = 2\pi r$ and $A = \pi r^2$. That's not a coincidence: it's because integrating circumference with respect to radius gives area. $\int_0^r 2\pi s \, ds = \pi r^2$. Onion-skin layers of circumference, summed up, fill the disk.
5. Arcs and sectors
Once you have the full circumference and area, taking a slice is a matter of proportion. An arc is some fraction of the circumference; a sector is the same fraction of the area. The fraction is determined by the central angle $\theta$ — the angle at the center between the two radii bounding the slice.
If $\theta$ is in degrees, the slice's share of the full $360°$ is $\theta/360°$. So:
$$ \text{arc length} \;=\; \frac{\theta}{360^\circ} \cdot 2\pi r $$ $$ \text{sector area} \;=\; \frac{\theta}{360^\circ} \cdot \pi r^2 $$The same two formulas in radians are cleaner — the conversion factor disappears, because radians were designed precisely so that $\theta = 1$ rad cuts off an arc of length $r$:
$$ s = r\theta \qquad A_\text{sector} = \tfrac{1}{2} r^2 \theta \quad (\theta \text{ in radians}) $$| Central angle $\theta$ | Fraction of circle | Arc length | Sector area |
|---|---|---|---|
| $90°$ (quarter) | $1/4$ | $\tfrac{1}{2}\pi r$ | $\tfrac{1}{4}\pi r^2$ |
| $180°$ (half) | $1/2$ | $\pi r$ | $\tfrac{1}{2}\pi r^2$ |
| $360°$ (full) | $1$ | $2\pi r$ | $\pi r^2$ |
Arc length is the distance along the curve, not the straight-line distance between the two endpoints. The straight segment is called the chord, and it's always shorter than the arc (except in the degenerate case where they coincide — which only happens when $\theta = 0$).
Central vs. inscribed angles
There are two ways to "measure" the arc cut off by two points $A$ and $B$ on a circle, and the distinction matters constantly in proofs.
- A central angle has its vertex at the center $O$ and sides along the radii $\overline{OA}$ and $\overline{OB}$. Its measure is, by definition, the measure of the arc it subtends — that's where the $\theta/360°$ fraction in the arc-length and sector formulas comes from.
- An inscribed angle has its vertex on the circle and sides along two chords $\overline{VA}$ and $\overline{VB}$ to the two endpoints of the arc.
The connecting fact — the inscribed angle theorem — is one of the cleanest results in all of geometry:
An inscribed angle is exactly half the central angle that subtends the same arc.
In symbols, if both angles cut off the same arc $AB$:
$$ \angle\text{inscribed} \;=\; \tfrac{1}{2} \, \angle\text{central} $$Two immediate corollaries fall out for free:
- Any inscribed angle that subtends a diameter (a $180°$ central angle) is a right angle. This is Thales' theorem — any triangle inscribed in a semicircle, with the diameter as one side, is automatically a right triangle.
- All inscribed angles subtending the same arc are equal to one another, regardless of where their vertex sits on the major arc — they all share the same central angle, so they all share half of it.
6. Tangents and secants
A straight line can interact with a circle in exactly three ways: miss it entirely, graze it at one point, or cut clean through. The last two have names.
- A tangent line touches the circle at exactly one point — the point of tangency. It runs alongside the curve without crossing it.
- A secant line intersects the circle at two points. The piece of the secant inside the circle is a chord; the rest extends out to infinity in both directions.
The defining property of a tangent — and the fact that gets used constantly — is that it is perpendicular to the radius drawn to the point of tangency:
$$ \text{tangent at } P \;\perp\; \overline{OP} $$This is the hinge for half the proofs in circle geometry. Once you know a line touches a circle at $P$, the radius $OP$ gives you a guaranteed right angle to work with.
The chord cut out by a secant — the segment $\overline{AB}$ above — and the various angle relationships between tangents, secants, and chords (the "two-secants theorem", the "tangent-secant angle", and so on) are where circle geometry starts to get genuinely interesting. Those build on the perpendicularity rule above.