Radius
GeometryThe 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 the edge. Every radius in the same circle is the same length. The diameter is exactly twice the radius.
Example
If a circle has a diameter of $10$ cm, the radius is $5$ cm. A bicycle wheel with a radius of $30$ cm has a diameter of $60$ cm. The radius is like the "arm" reaching from the center to the edge.
Key Insight
The word "radius" comes from Latin meaning "ray" or "spoke of a wheel." All radii of a circle are equal - that is what makes a circle so perfectly round. Change the radius and you change the size of the circle.
Definition
The radius of a circle is the constant distance from the center to any point on the circle. For a circle with radius $r$: circumference $= 2\pi r$, area $= \pi r^2$, diameter $d = 2r$. A tangent drawn from an external point $P$ at distance $d$ from center has length $\sqrt{d^2 - r^2}$.
Example
Circle with radius $7$: circumference $= 14\pi$ approximately $43.98$, area $= 49\pi$ approximately $153.94$. If a point $P$ is $25$ units from the center, the tangent length from $P$ to the circle $= \sqrt{625 - 49} = \sqrt{576} = 24$.
Key Insight
The tangent-radius theorem states that a radius to a point of tangency is perpendicular to the tangent line. This perpendicularity is why the tangent length formula involves the Pythagorean theorem: the radius, tangent, and center-to-point distance form a right triangle.
Definition
The radius $r$ of a circle determines it up to position (center). In the formula for curvature of a circle, $\kappa = 1/r$: a larger circle has smaller curvature. The circumradius $R$ of a triangle with sides $a,b,c$ and area $A$ is $R = abc/(4A)$. The inradius $r$ of a triangle is $r = A/s$ where $s$ is the semiperimeter.
Example
For a $3$-$4$-$5$ right triangle: $A=6$, $a=3$, $b=4$, $c=5$, $s=6$. Circumradius $R=60/24=2.5$ (equal to half the hypotenuse, confirming Thales' theorem). Inradius $r=6/6=1$. Check: $r=A/s=6/6=1$.
Key Insight
The Euler formula $R \ge 2r$ (circumradius at least twice inradius, with equality iff equilateral) elegantly captures the relationship between a triangle's two most important circle radii. Combined with $OI^2 = R(R-2r)$ (Euler's formula for the distance between circumcenter $O$ and incenter $I$), it forms a complete theory of triangle-circle relationships.