Radius
The radius is the distance from the center of a circle to any point on the circle, equal to half the diameter.
Formula
r = d/2; \text{Circumference} = 2\pi r; \text{Area} = \pi r^2
Definition
The radius is the distance from the center of a circle to any point on its edge; every radius in the same circle is equal, and the diameter is always exactly twice the radius, $d=2r$. For a circle of radius $r$: circumference $=2\pi r$, area $=\pi r^2$, and a tangent from an external point at distance $d$ from the center has length $\sqrt{d^2-r^2}$, a consequence of the tangent-radius theorem, since a radius to the point of tangency is always perpendicular to the tangent. The circumradius of a triangle with sides $a,b,c$ and area $A$ is $R=abc/(4A)$, and its inradius is $r=A/s$ (with $s$ the semiperimeter).
Example
A bicycle wheel with radius $30$ cm has diameter $60$ cm; if a circle has diameter $10$ cm, its radius is $5$ cm. For radius $7$: circumference $=14\pi\approx43.98$, area $=49\pi\approx153.94$; if an external point is $25$ units from the center, the tangent length is $\sqrt{625-49}=\sqrt{576}=24$. For a $3$-$4$-$5$ right triangle: circumradius $R=60/24=2.5$ (half the hypotenuse, confirming Thales' theorem) and inradius $r=6/6=1$.
Key Insight
The word "radius" comes from Latin meaning "ray" or "spoke of a wheel," and all radii of a circle being equal is exactly what makes it so perfectly round. The curvature of a circle is $\kappa=1/r$, so a larger circle has smaller curvature, and Euler's inequality $R\ge2r$ (with equality only for equilateral triangles) elegantly captures the relationship between a triangle's circumradius and inradius; combined with Euler's formula $OI^2=R(R-2r)$ for the distance between circumcenter and incenter, it forms a complete theory of triangle-circle relationships.