Comprehensive Calculator Guide
The distance between two points on a plane is the square root of the squared x-difference plus the squared y-difference: d = √((x₂ − x₁)² + (y₂ − y₁)²). From (0, 0) to (3, 4) it is 5, with the midpoint at (1.5, 2). Enter both points to get the distance and midpoint instantly.
Pythagoras in coordinates
The horizontal and vertical gaps between the points are the legs of a right triangle, and the distance is its hypotenuse. For (−1, 2) and (5, −6) the gaps are 6 and −8, their squares 36 and 64, and √100 = 10. Squaring removes the signs, so the order of points does not matter.
The midpoint averages the coordinates: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2), which for (2, 1) and (7, 13) is (4.5, 7). It also gives a circle's centre from the ends of a diameter.
Limits
In three dimensions add the third difference: √(Δx² + Δy² + Δz²). Latitude and longitude are on a sphere, so city-to-city distances need a spherical formula. The result here is a straight line, shorter than any real route.
How to Use
- 01Enter x₁ and y₁ for the first point.
- 02Enter x₂ and y₂ for the second; negatives are fine.
- 03Read the distance and midpoint, in the same unit as your coordinates.
Formula Used
d = √((x₂ − x₁)² + (y₂ − y₁)²) | midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)Practical Examples
Example 1: 3-4-5
(0, 0) and (3, 4): distance 5, midpoint (1.5, 2).
Example 2: negative coordinates
(−1, 2) and (5, −6): distance 10, midpoint (2, −2).
Example 3: a site plan in metres
(2, 1) and (7, 13): 13 m apart, midpoint (4.5, 7).
Important Tips
- The distance is never less than the larger gap nor more than the two gaps added.
- On a horizontal or vertical line, the distance is just the absolute difference.
Common Mistakes to Avoid
- Adding the gaps without squaring them.
- Using the formula on latitude and longitude.
Frequently Asked Questions
- What is the distance formula?
- d = √((x₂ − x₁)² + (y₂ − y₁)²), Pythagoras written in coordinates.
- How do I find the midpoint?
- Average the x-values and the y-values.
- Does the order of points matter?
- No; the differences are squared.
- Can I use it between two cities?
- No. Use a spherical distance formula or a mapping service.