The Distance Formula: Calculator, Proof & Worked Examples

By the Solve Formulas Editorial Team. Published June 12, 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?

The distance formula calculates the straight-line 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 most often: how far apart are these two points?

Subtract the x-coordinates, subtract the y-coordinates, square both results, add them, and take the square root. That is the full distance formula.

The distance formula and the Pythagorean theorem

You do not 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 do not need absolute-value bars inside the formula.

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 is a right triangle.

When to use the distance formula

You will use it whenever you have two coordinate pairs and need the length between them: checking if 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.

It works with negative numbers, decimals, and fractions. Subtract carefully; squaring takes care of the signs.

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 $d$.

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 $d = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07$.

Where you will see this in real life

• Game development. Engines use vector distance (same Pythagorean form) for collision checks and pathfinding on a 2D map.
• Data science. K-nearest neighbors and clustering measure how far apart data points are in feature space.
• Class and tests. SAT and ACT geometry often give coordinate vertices and ask about side lengths or right triangles.

GPS apps use spherical formulas such as the haversine equation, not 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.

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.

Common mistakes to avoid

• Forgetting to square before adding. If you got something like 7, you probably added $3 + 4$ instead of $9 + 16$.
• Mishandling negatives. $3 - (-2) = 5$, not −1.
• Taking the square root too early. Add the squared terms first, then take one square root.
• Mixing x and y. Always subtract x from x and y from y.

Rule of thumb: distance should never be negative. If your result is negative, re-check the arithmetic inside the radical before you took the root.

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