Is there a formula containing index of π (exclude index 1)

Question

By the way, someone help me to put it into a correct category?

I know following formula containing π (3.1415926....)

• Area of Circle: π * r * r
• Area of sphere: 4 * π * r * r
• Volume of sphere: 4/3 * π * r * r * r
• I've also seen π exist in some Trigonometric formula.

But I've never seen π is used like this:

• π^2
• π^(1/2)

Question 1: Is there any formula that involving π's index?

Question 2: If my guessing "no, there isn't any" is correct, is it because this mathematical constant is irrational or because it is transcendental or both or other feature?

Question 3: How about "Natural logarithm"? Any formula involving e's index?

Answer 1

• Volume of a $5$-dimensional ball: $\frac8{15}\pi^2 r^5$.
• Volume of a $6$-dimensional ball: $\frac16\pi^3 r^6$.
• Generalisation.
• $1+\frac1{16}+\frac1{81}+\frac1{256}+\cdots+\frac1{n^4}+\cdots=\frac1{90}\pi^4$.
• $\int_{-\infty}^\infty e^{-x^2}\,dx=\pi^{1/2}$.
• Many more!!

Answer 2

Very important in probability:

$$\int_{-\infty}^\infty e^{-x^2}\; dx = \sqrt{\pi}$$