# When is a triangle like a circle?

For a circle with radius $r$, its area is $\pi r^2$ and its circumference is $2\pi r$. If you take the derivative of the area formula with respect to $r$, you get the circumference formula! 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 $r^2$ 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 $s^2$, or $4r^2$, and its perimeter is $4s$, or $8r$. Sure enough, $dA/dr = d(4r^2)/dr = 8r = P$. What is the differential radius of an equilateral triangle with side length s?

Extra credit:
What is the differential radius of a rectangle with sides of length $a$ and $b$?

My solution:
[Show Solution]

## 2 thoughts on “When is a triangle like a circle?”

1. Mike Strong says:

Thanks for the great general solution!
I’m not sure that r = s*sqrt(3)/2 is correct for a triangle, though.
I came up with r = s*sqrt(3)/6,
and if you plug n= 3 into r = (s/2)*cot(pi/n), I think you get
r = (s/2)*cot(pi/3) = (s/2)*sqrt(3)/3) = s*sqrt(3)/6
Also, sqrt(3)/2 > 1, and it seems that the apothem for an equilateral triangle should be < 1.

1. Thanks for catching that! I meant to write $\frac{1}{2\sqrt{3}}$ but wrote $\frac{\sqrt{3}}{2}$ by accident. I fixed it now.