In high school I was always taught to use Pythagorean theorem to calculate distance or the distance formulas were always some variation of it. I've never seen a distance formula not using it. I can also see why it's useful because I think that if you have two points $(x_1,…,x_n)$ and $(y_1,…,y_n)$ in $\mathbb{R}^n$, you can use Pythagorean theorem $n−1$ times to show that the distance between them is $$\sqrt{\sum_{i=1}^{n}(x_i – y_i)^2}$$ So another question would be if there are other distance formulas why aren't they used?
[Math] way to calculate the distance between two points without using Pythagorean theorem
geometry
Best Answer
There are other distance formulas. The more technical term is a metric. One alternative is the Manhattan metric, also called the taxicab metric. For it we have $d(x,y)=\sum_i |x_i-y_i|$ with the motivation that if you are traveling in a city laid out in blocks the distance from one point to another is the number of blocks north/south plus the number of blocks east/west. Another is the discrete metric, where the distance between any two different points is $1$.