Mathematical problem

[Lestat]

Wolfram? theres an equation solver online?

There was me thinking it was only good for making armor-piercing ammunition, lightbulb filaments, and really bloody heavy stuff. (wolfram being a foreign name for tungsten)       

[Calavera]

I think I may have solved it. Hold on. There's a neat trick.

[Calavera]

Step 1 (rewriting original equation for clarification):
A = Pi * r * sqrt(r² + h²)

Step 2 (squaring both sides is safe because both sides are definitely positive):
A² = Pi² * r² * (r² + h²)

Step 3 (switch both sides around and then subtract A² from both sides to get a zero on the right side):
Pi² * r² * (r² + h²) - A² = 0

Step 4 (time for some expanding):
Pi² * r⁴ + Pi² * r² * h² - A² = 0

Step 5 (neat trick: adjust the equation so that it resembles a quadratic equation with r as the main variable):
Pi² * r² * r² + Pi² * h² * r * r - A² = 0

Step 6 (now focusing on the discriminant and fixing it a bit to make life easier):
discriminant = Pi⁴ * h⁴ * r² + 4 * Pi² * r² * A²

Step 7 (factoring the discriminant):
discriminant = Pi² * r² * (Pi² * h⁴ + 4 * A²)

Step 8 (pull one of the factors of the discriminant out of the square root sign):
discriminant = Pi * r * sqrt(Pi² * h⁴ + 4 * A²)

Step 9 (back to the bigger picture and solving for r using the infamous quadratic formula, rejecting the negative square root option because r must be positive):
r = [-(Pi² * h² * r) + discriminant] / (2 * Pi² * r²)

Step 10 (focusing now on the numerator of the fraction on the right side and switching the addition operands around thanks to the valuable property of commutativity):
numerator = discriminant - Pi² * h² * r

Step 11 (replace discriminant with its expression value and factor out the common factors):
numerator = Pi * r * (sqrt(Pi² * h⁴ + 4 * A²) - Pi * h²)

Step 12 (back to the overall equation and simplifying both the numerator and the denominator of the fraction on the right side of the equation):
r = (sqrt(Pi² * h⁴ + 4 * A²) - Pi * h²) / (2 * Pi * r)

Step 13 (multiplying both sides by r to get rid of the r on the right side):
r² = (sqrt(Pi² * h⁴ + 4 * A²) - Pi * h²) / (2 * Pi)

Step 14 (square root both sides to get the solution):
r = sqrt[(sqrt(Pi² * h⁴ + 4 * A²) - Pi * h²) / (2 * Pi)]

And voila! Am I the king of the universe or what?