How can you calculate the distance between 2 points in a grid if they're not on the same horizontal or vertical line?

Recall Pythagoras' theorem. a^2 + b^2 = c^2. Think about this as a triangle, with a and b being the lengths of the 2 perpendicular sides, and c being the length of the hypotenuse. [Drawing on a whiteboard would help!:P]

If we want to find the distance between the points P1 and P2 [draws points on board], we can draw the line between them and make that the hypotenuse of the triangle. Now we'll fill in the sides of length a and b to show the full triangle.

We can find out the lengths a and b with simple subtractions: P2.x - P1.x = a, for example. Now, we can find c^2 with pythagoras: a^2 + b^2. Just plug in the values of P1.x, P1.y, P2.x, P2.y we had earlier: (P2.x - P1.x)^2 + (P2.y - P1.y)^2 = c^2 This might look quite scary, but you can see exactly how we got here. Remember, c is the distance between the two points. To find c, we just need to square root c^2, so our final expression for c becomes sqrt((P2.x - P1.x)^2 + (P2.y - P1.y)^2)

[This is very awkward to type, which is probably why it looks so bad:P]