Circle
A 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, the radius, from the center. Circumference $C=2\pi r=\pi d$, area $A=\pi r^2$, and the equation of a circle centered at $(h,k)$ is $(x-h)^2+(y-k)^2=r^2$, essentially the Pythagorean theorem applied to the distance from any point $(x,y)$ to the center. Formally, a circle in $\mathbb{R}^2$ is a $1$-sphere $S^1$, a conic section (a cone sliced parallel to its base), and by the isoperimetric inequality, among all closed curves of fixed length $L$, the circle encloses the maximum possible area $A=L^2/(4\pi)$.
Example
A coin, a wheel, and the top of a can are all circles; a circle with radius $3$ cm has circumference $2\pi\cdot3\approx18.8$ cm and area $9\pi\approx28.3$ square cm. A circle centered at $(3,-2)$ with radius $5$ has equation $(x-3)^2+(y+2)^2=25$, circumference $10\pi\approx31.4$, area $25\pi\approx78.5$. A circle of circumference $10$ encloses area $10^2/(4\pi)=25/\pi\approx7.96$, the maximum possible area for that perimeter, with equality holding only for a true circle.
Key Insight
Pi (approximately $3.14159$) is the ratio of a circle's circumference to its diameter, the same for every circle regardless of size, which is what makes $\pi$ so universal. The isoperimetric inequality, that a circle maximizes area for a given perimeter, is one of the oldest extremal results in mathematics, known to the ancient Greeks though its rigorous proof required 19th-century analysis, and it generalizes to $n$-spheres and to Riemannian manifolds.