# Approximate that hypotenuse

Ever found yourself wanting to approximate a hypotenuse quickly? Probably the most common situation where this happens, at least in my experience, is when you want to know the range between two combatants on a battle map when one is at a different altitude. You don’t always want to use a calculator in play, so what do you do instead? Is there a way to quickly approximate √x2 + y2 in play?

I’ve thought of and used a couple different approximations over the years, but yesterday I thought up one that’s really quite good:

For two numbers X and Y, where X ≥ Y, ((X * 2) + Y)/2 ≈ √x2 + y2

That is, add X and X and Y, then divide the result by 2. So if you’re trying to find the hypotenuse of 3 and 4, this formula gives an approximation of ((4 * 2) + 3)/2, or 5.5, and the actual answer is 5. Not bad for an on-the-fly answer.

Matt pointed out that if X ≥ 2Y, it’s actually more accurate to just use X straight up as the hypotenuse. So if X is 20 and Y is 5, the formula’s approximation would be 22.5, the actual hypotenuse would be 20.615…, and the approximation would be 20. Even better.

If you end up using this in a game, let me know!