This week the question is: When Is a Triangle Like a Circle?

… let’s define the term “differential radius.” The differential radius r of a shape with area A and perimeter P (both functions of r) has the property that dA/dr = P. (Note that A always scales with r2 and P always scales with r.)

For example, consider a square with side length s. Its differential radius is r = s/2. The square’s area is s2, or 4r2, and its perimeter is 4s, or 8r. Sure enough, dA/dr = d(4r2)/dr = 8r = P.

What is the differential radius of an equilateral triangle with side length s?

Highlight to reveal solution:

I pretty much just worked through equations for equilateral triangles. I also assumed r = a * s. In other words, a simple relationship. After some calculations, I got r = s / (2 * √3).