Question

What is the Radius of a circle that has the same numerical value for area and circumference? How do the units compare?

Answer 1

The area of a circle is given by

$A = \pi r^2$

whereas the circumference is given by

$C = 2\pi r$

If we want these two values to be (numerically) the same we have to set $A=C$ and solve for the radius:

$A = C \iff \pi r^2 = 2\pi r \iff r^2 = 2r \iff r^2-2r=0 \iff r(r-2) = 0$

So, one (trivial) solution is $r=0$. A circle with radius 0 is just a point, and so both area and circumference are zero.

The other solution is $r = 2$. In fact, you have

$A = \pi r^2 = 4\pi,\quad C = 2\pi r = 2\pi \cdot 2 = 4\pi$