Distance Formula

1. Introduction: Finding how far apart two points are — without a ruler

Here is the idea. You have two points on a graph. You want the straight-line distance between them. Not by measuring with a ruler. By calculating it from their coordinates.

A quick reminder first. A point's coordinates are the two numbers that fix its position, written as $(x, y)$. The first number $x$ tells you how far across (left-right). The second number $y$ tells you how far up-down. So $A(4, 6)$ means: go 4 across, 6 up.

Why do we need this at all? Picture a town B that sits 36 km east and 15 km north of town A. You want the distance from A to B as the crow flies — the straight path, not along the roads. You cannot walk it with a measuring tape. But you know the two numbers: 36 east, 15 north. From just those two numbers, you can find the distance.

By the end of this lesson, you will have one formula that does this for any two points, anywhere on the graph.

Stop scrolling. Try it in your head before reading on: town A to town B, 36 km east and 15 km north — does that "36 east, 15 north" remind you of any shape you have seen?