Distance Formula: Calculator, Step-by-Step Proof & Examples

By Tim Carter, Math Educator & Founder · Published June 12, 2026 · Updated June 15, 2026

Why this calculator exists. Most online tools give you the answer and nothing else. We built this one to show every step, because on a test, knowing how you got $d = 5$ is the whole point.

What is the distance formula?

By definition, the distance formula calculates the straight-line (Euclidean) distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on a coordinate plane. It answers the question students ask in every geometry class: how far apart are these two points? The distance between two points formula gives you a single number: the length of the segment connecting them.

Subtract the x-coordinates, subtract the y-coordinates, square both results, add them, and take the square root. That's the full distance formula: five steps, every time.

Distance formula proof: the connection to the Pythagorean theorem

You don't need a separate rule to memorize. Plot the two points, draw a right triangle whose hypotenuse connects them, and the legs are $|x_2 - x_1|$ and $|y_2 - y_1|$. The Pythagorean theorem gives:

Take the positive square root and you have the distance formula. Squaring removes the sign, so you don't need absolute-value bars inside the formula, so it works with negative coordinates automatically.

SAT tip. When a problem gives you three coordinate pairs, find all three side lengths with the distance formula first. Then check whether the longest side squared equals the sum of the other two squared. Example: vertices $(0,0)$, $(3,0)$, $(0,4)$ give sides 3, 4, and 5. Since $3^2 + 4^2 = 5^2$, it's a right triangle.

When to use the distance formula

Use it whenever you need the length between two coordinate pairs, or put another way, whenever you need to find the distance between coordinates. Common situations: checking whether a triangle is isosceles (compute all three sides and compare), finding a circle's radius from its center to a point on the edge, or getting the diagonal of a rectangle on a grid.

Negative numbers, decimals, fractions: it handles all of them. Subtract carefully; squaring fixes the signs after that.

How to find the distance between two points (step by step)

1. Label your points as $(x_1, y_1)$ and $(x_2, y_2)$.
2. Subtract: compute $x_2 - x_1$ and $y_2 - y_1$.
3. Square each difference.
4. Add the two squares.
5. Take the square root of the sum. That value is $d$.

The most common mistake happens at step 3: students sometimes add the differences before squaring. Always square first, then add.

Walk-through with $(0, 0)$ and $(3, 4)$:

Distance formula examples

Example 1: Whole-number coordinates (3-4-5 triangle)

Distance between $A(1, 2)$ and $B(4, 6)$:

Classic 3-4-5 right triangle. The answer is exactly 5 units.

Example 2: Negative coordinates

Between $P(-2, -1)$ and $Q(3, 11)$:

Example 3: Decimal coordinates

Between $(0.5, 1.5)$ and $(3.25, 4.75)$:

Example 4: Distance from the origin

Between $O(0, 0)$ and $(6, 8)$:

When one point is the origin, the formula simplifies to $d = \sqrt{x^2 + y^2}$.

3D distance formula

In three dimensions, add a $z$ term under the root for points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$:

Example: $P(1, 2, 3)$ to $Q(4, 6, 8)$ gives $d = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07$. Same formula, one extra leg.

Distance formula and Pythagorean theorem: real-life applications

• Game development. Engines use vector distance (the same Pythagorean form) for collision detection and pathfinding on a 2D map. Two characters with positions $(10, 20)$ and $(13, 24)$ are $\sqrt{9+16} = 5$ units apart.
• Construction. A carpenter cutting a diagonal brace for a rectangular frame measuring 3 ft by 4 ft uses $\sqrt{3^2 + 4^2} = 5$ft for the brace length. That's the distance formula (and the Pythagorean theorem) in action.
• Data science. K-nearest neighbors and clustering algorithms measure how far apart data points are in feature space using Euclidean distance, the same formula extended to many dimensions.
• Tests. SAT and ACT geometry questions often give coordinate vertices and ask about side lengths, right triangles, or circle radii.

GPS apps use spherical formulas such as the haversine equation rather than flat Euclidean distance, because the Earth is curved. For a homework grid, Euclidean distance is what you want.

Distance formula practice problems

Try these, then open the answers to check your work. If you want more practice, treat the five problems below as a mini distance formula worksheet: write them out by hand before checking each answer.

1. Find $d$ between $(0, 0)$ and $(3, 4)$.
2. Find $d$ between $(-1, 2)$ and $(5, -3)$.
3. Find $d$ between $(\frac{1}{2}, \frac{1}{3})$ and $(\frac{3}{2}, \frac{5}{3})$.
4. In 3D, find $d$ between $(1, 1, 1)$ and $(4, 5, 6)$.
5. Triangle $ABC$ has vertices $A(0,0)$, $B(4,0)$, $C(2,3)$. Find each side length and identify the triangle type.

Common mistakes to avoid

• Adding before squaring. If you got something like 7, you probably added $3 + 4$ instead of $9 + 16$. Always square first.
• Mishandling negatives. $3 - (-2) = 5$, not −1. Draw the subtraction out if you're unsure.
• Taking the square root too early. Add the squared terms first, then take one square root at the end.
• Mixing x and y. Always subtract x from x and y from y; keep the coordinates in their own columns.

Rule of thumb: distance should never be negative. If your result is negative, something went wrong before the square root step; check the arithmetic inside the radical.

Related geometry formulas

The distance formula is part of a cluster of coordinate geometry tools. Once you have the distance between two points, the midpoint formula tells you the coordinates of the point exactly halfway between them, useful for finding the center of a segment. The slope formula uses the same coordinate differences to describe the steepness of the line connecting the two points. These three formulas together cover the most common coordinate geometry questions on the SAT and ACT.

The distance formula also appears inside the equation of a circle: the standard form $(x - h)^2 + (y - k)^2 = r^2$ is simply the distance formula squared, where $(h, k)$ is the center and $r$ is the radius.

Explore more on the geometry formulas hub, or jump to the distance formula calculator at the top of this page.