A Distance Between 3 Points Calculator is a mathematical tool that helps you find the exact distances between three points in either a two-dimensional plane or three-dimensional space. This calculator takes the coordinates of three points and computes the distance between each pair of points.
The calculator performs the following functions:
- It accepts the coordinates of three points (either in 2D or 3D space).
- It applies the distance formula to calculate the distance between each pair of points.
- It provides the three resulting distances: between points 1 and 2, points 2 and 3, and points 3 and 1.
This tool is especially useful for students studying geometry, professionals working in fields like architecture, engineering, computer graphics, and anyone dealing with spatial calculations.
Formula of Distance Between 3 Points Calculator
To calculate the distance between three points in a three-dimensional space, the distances between each pair of points must be determined using the distance formula.
The formula for the distance between two points (x1, y1, z1) and (x2, y2, z2) is:
d = sqrt((x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2)
For three points, labeled as (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), the distances are:
d12 = sqrt((x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2)
d23 = sqrt((x3 – x2)^2 + (y3 – y2)^2 + (z3 – z2)^2)
d31 = sqrt((x1 – x3)^2 + (y1 – y3)^2 + (z1 – z3)^2)
These values represent the distances between the three points in space.
For two-dimensional calculations (when all points lie on a plane), the formula simplifies to:
d = sqrt((x2 – x1)^2 + (y2 – y1)^2)
And the three distances would be:
d12 = sqrt((x2 – x1)^2 + (y2 – y1)^2)
d23 = sqrt((x3 – x2)^2 + (y3 – y2)^2)
d31 = sqrt((x1 – x3)^2 + (y1 – y3)^2)
Common Distance Values Reference Table
Here’s a handy reference table for some common coordinate patterns and their resulting distances:
| Point 1 (x,y,z) | Point 2 (x,y,z) | Point 3 (x,y,z) | d12 | d23 | d31 |
|---|---|---|---|---|---|
| (0,0,0) | (1,0,0) | (0,1,0) | 1 | 1.414 | 1 |
| (0,0,0) | (1,1,1) | (2,2,2) | 1.732 | 1.732 | 3.464 |
| (0,0,0) | (3,0,0) | (0,4,0) | 3 | 5 | 4 |
| (1,1,1) | (2,2,2) | (3,3,3) | 1.732 | 1.732 | 3.464 |
| (0,0,0) | (5,5,0) | (10,0,0) | 7.071 | 7.071 | 10 |
| (0,0,0) | (0,0,5) | (5,5,5) | 5 | 7.071 | 8.660 |
| (1,2,3) | (4,5,6) | (7,8,9) | 5.196 | 5.196 | 10.392 |
| (0,0,0) | (10,0,0) | (0,10,0) | 10 | 14.142 | 10 |
This table shows common coordinate sets and their corresponding distances. The values are rounded to three decimal places for simplicity.
Example of Distance Between 3 Points Calculator
Let’s work through a practical example to understand how to calculate the distance between three points.
Suppose we have three points in 3D space:
- Point A: (1, 2, 3)
- Point B: (4, 5, 6)
- Point C: (7, 8, 1)
To find the distances between these points:
- Distance between A and B (d12):
d12 = sqrt((4-1)^2 + (5-2)^2 + (6-3)^2)
d12 = sqrt(9 + 9 + 9)
d12 = sqrt(27)
d12 = 5.196
- Distance between B and C (d23):
d23 = sqrt((7-4)^2 + (8-5)^2 + (1-6)^2)
d23 = sqrt(9 + 9 + 25)
d23 = sqrt(43)
d23 = 6.557
- Distance between C and A (d31):
d31 = sqrt((1-7)^2 + (2-8)^2 + (3-1)^2)
d31 = sqrt(36 + 36 + 4)
d31 = sqrt(76)
d31 = 8.718
Therefore, the distances between our three points are approximately:
- A to B: 5.196 units
- B to C: 6.557 units
- C to A: 8.718 units
These calculations can help you understand the spatial relationship between the three points and can be useful for various applications like determining if the points form a right triangle, calculating the perimeter of the triangle formed, or finding the area using Heron’s formula.
Most Common FAQs
The distance between three points refers to three separate measurements: the distance from point 1 to point 2, from point 2 to point 3, and from point 3 to point 1. Each uses the same distance formula for two points, effectively measuring the three sides of the triangle formed by these points.
Yes! Use the Pythagorean theorem: if the three distances (a, b, c) satisfy a² + b² = c² (where c is the longest distance), the points form a right triangle. Example: distances of 3, 4, and 5 units form a right triangle because 3² + 4² = 5².
Simply add the three distances: Perimeter = d12 + d23 + d31. For distances of 5, 6, and 7 units, the perimeter is 5 + 6 + 7 = 18 units.