Radius

The pair

$$d = 2r \qquad r = \tfrac{d}{2}$$

• radius — centre to edge
• diameter — right across, through the centre

Every radius of a circle is the same length. That is what a circle is: all the points a fixed distance from one centre.

Which one a formula wants

Getting this wrong is the dominant error in circle work, and each formula fails in its own recognisable way.

Formula	Wants	Wrong input gives
$C = \pi d$	diameter	a radius halves the answer
$C = 2\pi r$	radius	a diameter doubles it
$A = \pi r^2$	radius	a diameter quadruples it

The area case is worse than the others because the mistaken value is then squared, so doubling becomes quadrupling. An answer exactly four times the expected size is nearly always a diameter sitting where a radius belongs.

The plural

Two or more are radii, pronounced “ray-dee-eye”. Diameters is regular.

Beyond circles

A sphere has a radius too, and the same distinction applies to its formulas. See circumference of a circle and area of a circle.