Circle
GeometryA circle is the set of all points in a plane that are the same distance (the radius) from a fixed center point.
Formula
\text{Area} = \pi r^2; \text{Circumference} = 2\pi r
Definition
A circle is a perfectly round shape where every point on the edge is the same distance from the center. That distance is called the radius.
Example
A coin, a wheel, and the top of a can are all circles. If the radius is $3$ cm, the circumference (distance around) is $2\pi\cdot3$ approximately $18.8$ cm, and the area is $9\pi$ approximately $28.3$ square cm.
Key Insight
Pi (approximately $3.14159$) is the ratio of a circle's circumference to its diameter - the same for EVERY circle, no matter the size. That constant ratio is what makes $\pi$ so special and universal.
Definition
A circle is the set of all points in a plane equidistant from a fixed center point. The common distance is the radius $r$. Circumference $C = 2\pi r = \pi d$. Area $A = \pi r^2$. The equation of a circle centered at $(h, k)$ is $(x-h)^2 + (y-k)^2 = r^2$.
Example
Circle centered at $(3, -2)$ with radius $5$: equation $(x-3)^2 + (y+2)^2 = 25$. Circumference $= 10\pi$ approximately $31.4$. Area $= 25\pi$ approximately $78.5$. The point $(3+5, -2) = (8, -2)$ lies on the circle.
Key Insight
The equation $(x-h)^2 + (y-k)^2 = r^2$ is the Pythagorean theorem applied to the distance from $(x,y)$ to $(h,k)$. Every circle equation is essentially a distance formula set equal to a constant - connecting circles directly to the Pythagorean theorem.
Definition
A circle in $\mathbb{R}^2$ is a $1$-sphere $S^1$: the set $\{(x,y) : (x-h)^2 + (y-k)^2 = r^2\}$. It is a conic section (intersection of a cone with a plane parallel to the base). In complex analysis, the unit circle $|z| = 1$ is fundamental in the theory of analytic functions. The isoperimetric inequality states: among all closed curves of fixed length $L$, the circle encloses the maximum area $A = L^2/(4\pi)$.
Example
Isoperimetric inequality: a circle of circumference $10$ has area $10^2/(4\pi) = 25/\pi$ approximately $7.96$. Any other closed curve of circumference $10$ encloses less area. Equality holds iff the curve is a circle.
Key Insight
The isoperimetric inequality (circle maximizes area for a given perimeter) is one of the oldest extremal results in mathematics, known to ancient Greeks. Its rigorous proof required 19th-century analysis. The inequality generalizes to $n$ dimensions ($n$-sphere maximizes volume for given surface area) and to Riemannian manifolds.