Math Tools
Distance Calculator
Calculate the distance between two coordinate points using the distance formula.
Distance Calculator
Calculate the straight-line distance between two points in a 2D plane.
Formula
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Results
Distance
5
Δx
3
Δy
4
Substituted formula
d = √((4 - 1)² + (6 - 2)²)
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What this calculator does
This calculator finds the straight-line distance between two points in a 2D plane using the distance formula. It also shows the horizontal and vertical changes between the points.
What is a distance calculator?
A distance calculator is a tool used to find the straight-line distance between two points on a coordinate plane. By entering the coordinates of point one and point two, you can quickly calculate the exact distance without solving the equation by hand.
This is useful in geometry, algebra, graphing, physics, engineering, navigation, and many other subjects where you need the distance between two coordinates.
Distance formula between two points
The distance between two points in a 2D plane is calculated with the formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this formula, x₁ and y₁ are the coordinates of the first point, and x₂ and y₂ are the coordinates of the second point. The formula measures the straight-line distance between them.
Distance formula symbols and meanings
The table below explains the main parts of the distance between two points formula.
| Symbol | Meaning | Example |
|---|---|---|
| x₁, y₁ | Coordinates of the first point | (1, 2) |
| x₂, y₂ | Coordinates of the second point | (4, 6) |
| Δx | Horizontal change between the two x-values | 4 - 1 = 3 |
| Δy | Vertical change between the two y-values | 6 - 2 = 4 |
| d | Straight-line distance between the two points | √(3² + 4²) = 5 |
How to calculate the distance between two points step by step
First, subtract the x-coordinate of the first point from the x-coordinate of the second point to get Δx. Then subtract the y-coordinate of the first point from the y-coordinate of the second point to get Δy.
Next, square both values, add them together, and then take the square root of the total. The final result is the distance between the two points.
This method gives the direct distance, not the horizontal or vertical travel distance separately.
Distance formula examples
These worked examples show how to calculate distance, Δx, and Δy for common coordinate pairs.
| Points | Δx | Δy | Distance |
|---|---|---|---|
| (1, 2) to (4, 6) | 3 | 4 | 5 |
| (0, 0) to (3, 4) | 3 | 4 | 5 |
| (-2, 1) to (4, 5) | 6 | 4 | 7.211103 |
| (2, -3) to (7, 9) | 5 | 12 | 13 |
What do Δx and Δy mean?
Δx represents the horizontal change between the two points. It is found by subtracting x₁ from x₂. Δy represents the vertical change and is found by subtracting y₁ from y₂.
These two values are important because the distance formula is based on the Pythagorean theorem. The horizontal and vertical changes act like the two legs of a right triangle, and the distance is the length of the hypotenuse.
Distance formula and the Pythagorean theorem
The distance formula is a direct application of the Pythagorean theorem. When two coordinate points are plotted, the horizontal change and vertical change form the two legs of a right triangle.
The straight-line distance between the points is the hypotenuse of that triangle, which is why the formula uses squared differences and a square root.
Where a distance calculator is useful
A distance calculator is useful in school math, coordinate geometry, graphing, physics, engineering, and mapping problems. It helps you measure how far apart two points are in a direct straight line.
It can also be useful when checking plotted points, solving test problems, verifying homework, or understanding how coordinate changes affect total distance.
Distance calculator FAQ and common questions
How do you find the distance between two points?
Use the distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²). This gives the straight-line distance between the two coordinates.
What is the formula for distance in coordinate geometry?
The coordinate geometry distance formula is d = √((x₂ - x₁)² + (y₂ - y₁)²).
What does Δx mean in the distance formula?
Δx is the horizontal difference between the x-values of the two points. It is calculated as x₂ - x₁.
What does Δy mean in the distance formula?
Δy is the vertical difference between the y-values of the two points. It is calculated as y₂ - y₁.
Why does the distance formula use a square root?
The square root is used because the formula comes from the Pythagorean theorem. After squaring and adding the horizontal and vertical differences, the square root gives the actual distance.
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