Calculator Inputs
Interactive Plot
The chart plots both points and the segment connecting them.
Formula Used
The distance formula comes from the Pythagorean theorem. The horizontal change becomes one leg of a right triangle, and the vertical change becomes the other. Squaring both changes removes sign direction. Adding those squares and taking the square root gives the straight-line distance between two points.
When the value under the radical is not a perfect square, the exact answer remains in radical form. For example, √52 simplifies to 2√13 because 52 = 4 × 13.
How to Use This Calculator
- Enter labels for your two points if needed.
- Type x₁, y₁, x₂, and y₂ coordinates.
- Press Calculate Distance.
- Read the exact radical form and decimal result.
- Review midpoint, slope, and the full step breakdown.
- Use the CSV or PDF buttons to save your result.
Example Data Table
| Point A | Point B | Δx | Δy | Distance Squared | Exact Distance | Decimal Distance |
|---|---|---|---|---|---|---|
| (1, 2) | (4, 6) | 3 | 4 | 25 | 5 | 5 |
| (2, 3) | (8, 11) | 6 | 8 | 100 | 10 | 10 |
| (-1, 5) | (5, 8) | 6 | 3 | 45 | 3√5 | 6.708204 |
| (0, 0) | (7, 9) | 7 | 9 | 130 | √130 | 11.401754 |
Answers to Common Questions
How do you calculate distance between two GPS coordinates?
For GPS coordinates, use the haversine formula or spherical law of cosines because Earth is curved. Convert latitude and longitude from degrees to radians before computing great-circle distance.
What formula estimates the distance of a basketball launch to the basket?
If you only need straight-line separation, use the distance formula between release and basket coordinates. For travel path length of the shot, model projectile motion instead of a straight segment.
FAQs
1. What is the distance formula in coordinate geometry?
It finds the straight-line distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. It is derived directly from the Pythagorean theorem.
2. Why does this calculator show radicals?
Radicals preserve the exact form of the answer. A decimal may round the value, but a simplified radical like 3√5 stays mathematically exact.
3. When will the radical simplify?
The radical simplifies when the value under the square root contains a perfect-square factor. For example, √72 simplifies to 6√2 because 72 = 36 × 2.
4. Can I use negative coordinates?
Yes. Negative x-values and y-values work normally. Squaring the coordinate differences ensures the distance remains nonnegative.
5. What happens if both points are identical?
The horizontal and vertical changes are both zero, so the distance is zero. The midpoint is the same point, and the slope is undefined.
6. Is this tool for 2D points only?
Yes, this version is designed for two-dimensional coordinate geometry. A 3D version would also include z-values inside the square-root expression.
7. Why show both exact and decimal results?
Exact form is better for algebra and proofs. Decimal form is better for graphing, measurement, and practical estimation tasks.
8. Does the graph affect the calculation?
No. The graph is only a visual aid. The calculation comes from the coordinate inputs and the distance formula itself.