9 common myths about the distance formula debunked

The distance formula looks intimidating. It is packed with subscripts, negative signs, squares, and a massive square root symbol covering the whole thing. Because the equation looks messy, people tend to overcomplicate how it works.

Let’s clear up the confusion by walking through nine common myths about the distance formula, alongside the actual math.

Myth 1: The order in which you enter the points changes the answer

If you swap Point 1 and Point 2, will your final distance change? It feels like it should, because subtraction is directional. Subtracting 5 − 2 is not the same as 2 − 5.

But the distance formula squares the result of that subtraction, which irons out the negative signs. Squaring a negative number always gives you a positive one. Take Point A at (0, 0) and Point B at (3, 4). The horizontal difference is either (0 − 3) or (3 − 0).

Since (−3)² is 9 and 3² is 9, the order is irrelevant. The final distance stays exactly the same.

Myth 2: Negative coordinates break the formula

It is easy to look at negative coordinates—moving left or down on the grid—and assume you need a tweaked formula. You don’t. The standard equation absorbs negative numbers easily.

The minus signs built into the formula are just there to find the gap between two points. If you plug in a negative coordinate, you end up subtracting a negative. That is mathematically identical to adding. Say your x-coordinates are 2 and −5. The horizontal difference is 2 − (−5). This flips to 2 + 5, which equals 7. No special rules are required.

Myth 3: You can cancel out the squares and the square root

This is a classic algebraic trap. Because the formula wraps squared terms inside a giant square root, your brain desperately wants to cancel them out. It seems logical to assume the distance is simply the horizontal change plus the vertical change.

But mathematically, √(a² + b²) is never equal to a + b unless one of those variables is zero.

Using our points (0, 0) and (3, 4), the horizontal change is 3 and the vertical change is 4. If you drop the squares and the root, you add 3 + 4 to get 7. That is wrong. You have to square them first. The correct path is √(3² + 4²), which turns into √(9 + 16), simplifying to √25. The actual distance is 5.

Myth 4: The formula is unrelated to basic geometry

Written out as d = √((x₂ − x₁)² + (y₂ − y₁)²), it looks like pure algebra. But the distance formula is literally just the Pythagorean theorem (a² + b² = c²) applied to a coordinate grid.

If you draw a straight line between any two points, that line acts as the hypotenuse—the longest side—of a right triangle. The horizontal distance (dx) and the vertical distance (dy) are the two shorter legs of that triangle. So, dx² + dy² = distance². The formula just isolates the distance by taking the square root of both sides.

Myth 5: A calculated distance can be negative

Coordinates can be negative. The slope of a line can be negative. It stands to reason that distance could be negative, too.

Except it cannot. Distance is a magnitude, meaning it measures physical length. It is always a positive number, or exactly zero if your two points are sitting on the exact same spot. The square root in the formula asks for the principal, or positive, square root. Even if you trace a path backward along an axis, the length of the string connecting those points remains positive.

Myth 6: You must use the full formula for straight horizontal or vertical lines

If two points share an axis, running them through the full equation is a waste of time.

Say you want the distance between (2, 5) and (8, 5). The y-coordinates are identical, meaning the line is perfectly flat. The vertical difference is zero, so the formula collapses to the square root of the horizontal difference squared. Skip the heavy math. Just find the absolute difference between the numbers that actually change. The gap between 2 and 8 is 6. You are done.

Myth 7: The midpoint formula relies on subtraction

Because students usually learn the midpoint formula alongside the distance formula, the wires often get crossed. People try to find the midpoint by subtracting coordinates.

Distance measures the gap between points, which requires subtraction. Midpoint finds the exact center location. To find a center, you need an average. You find it by adding the two x-coordinates and dividing by two, then repeating that for the y-coordinates. For (0, 0) and (3, 4), the midpoint is (1.5, 2).

Myth 8: A slope of zero means the distance is zero

Slope and distance measure entirely different properties. Slope dictates steepness and direction. Distance dictates length.

A line with a slope of zero is simply flat. The distance is just the horizontal length of that flat line.

On the flip side, a perfectly vertical line has a horizontal change (dx) of zero. Calculating slope requires dividing by dx, and dividing by zero is undefined. So a vertical line has an undefined slope, but its distance is still perfectly measurable—it is simply the vertical length between the points.

Myth 9: The formula only works on a 2D coordinate plane

We learn this formula using an x-axis and a y-axis, which leaves the impression that it only applies to flat, two-dimensional surfaces.

But Euclidean distance scales easily into three-dimensional space. To find the distance between two points in a 3D environment, you just extend the Pythagorean logic. Find the difference in the z-axis, square it, and tuck it under the same square root next to the x and y values. The math expands to handle extra dimensions without altering the core mechanics.

Ready to find the exact length, midpoint, and slope between your coordinates? Use our Distance Formula Calculator.