Is $\frac{\text{area of circle with diameter $d$}}{\text{area of square with side $d$}}$ irrational?

Question

I understand that the area of a circle (based on diameter) with respect to the area of a square (same side length as circle diameter) is approximately 0.7854 the area of the square. My question:

Is $$\frac{\text{area of circle with diameter $d$}}{\text{area of square with side $d$}}$$ irrational?

Answer 1

It's $\frac\pi4$, and since $\pi$ is transcendental – much, much stronger than irrational – so is $\frac\pi4$.

Answer 2

First of all, the radius of the circle is $\frac{d}{2}$. This gives the area of the circle = $(\frac{d}{2})^2*\pi = \frac{\pi d^2}{4}$. The area of the square with side $d$ is just $d^2$. So, the fraction you ask for is $\frac{\frac{\pi d^2}{4}}{d^2} = \frac{\pi}{4}.$ Since $\pi$ is irrational, $\frac{\pi}{4}$ is also irrational.