Distance Calculator
Use this Distance Calculator to find distance in several smart ways: between two points, in 3D, from a point to a line, with speed, time, and distance, using latitude and longitude, or on a simple scaled coordinate grid. It is built to be educational, user friendly, and helpful for both math problems and real-world distance questions.
Background
In math, distance measures how far apart two objects are. In coordinate geometry, distance often comes from the Pythagorean theorem. In real-world travel and navigation, distance may be measured on a flat grid, along a route, or across the Earth using great-circle distance. This calculator brings these ideas together in one flexible tool.
How to use this calculator
- Choose the distance mode that matches your problem.
- Enter the required values, such as coordinates, line coefficients, or speed and time values.
- Click Calculate to see the answer, supporting values, a visual, and optional step-by-step work.
How this calculator works
- For two points in 2D, it uses d = √((x₂ − x₁)² + (y₂ − y₁)²).
- For 3D points, it uses d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).
- For a point to a line, it uses d = |Ax₀ + By₀ + C| / √(A² + B²).
- For speed, time, and distance, it uses d = rt, r = d/t, or t = d/r.
- For latitude and longitude, it uses the Haversine formula to estimate great-circle distance on Earth.
- For scaled grid mode, it first finds straight-line grid distance, then applies your chosen scale.
Formulas & Equations Used
2D distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
3D distance: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
Point to line: d = |Ax₀ + By₀ + C| / √(A² + B²)
Distance / speed / time: d = rt
Haversine: a = sin²(Δφ/2) + cos(φ₁)cos(φ₂)sin²(Δλ/2)
Great-circle distance: d = 2R · asin(√a)
Example Problems & Step-by-Step Solutions
Example 1 — Distance between two points
Find the distance between (1, 2) and (4, 6).
- Find the horizontal change: 4 − 1 = 3.
- Find the vertical change: 6 − 2 = 4.
- Use the distance formula: d = √(3² + 4²) = √25 = 5.
Example 2 — Point to line distance
Find the distance from (3, 4) to the line 2x − y − 5 = 0.
- Substitute the point into the numerator: |2(3) + (-1)(4) + (-5)| = |6 - 4 - 5| = 3.
- Find the denominator: √(2² + (-1)²) = √5.
- Distance is 3 / √5.
Example 3 — Latitude and longitude distance
Enter two coordinate pairs and the calculator estimates the shortest path along Earth’s surface in kilometers, miles, and nautical miles.
Frequently Asked Questions
Q: What is the distance formula?
The standard 2D distance formula is d = √((x₂ − x₁)² + (y₂ − y₁)²).
Q: Can distance be negative?
No. Distance is a length, so it is always zero or positive.
Q: What is the difference between straight-line distance and scaled grid distance?
Straight-line distance is the shortest path between two points. Scaled grid distance uses that straight-line result and applies a chosen scale. It is not the same as route or driving distance.
Q: What does latitude and longitude distance measure?
It estimates the shortest distance over the Earth’s surface between two geographic coordinates.
