3D Distance Calculator (Distance Between Two Points in Space from xyz Coordinates)

Enter the xyz coordinates of point 1 and point 2 to find the distance between them in 3D space. The changes in x, y and z are shown too.

Point 1 x₁

Point 1 y₁

Point 1 z₁

Point 2 x₂

Point 2 y₂

Point 2 z₂

Enter the coordinates as numbers (decimals and negative numbers are OK). All six fields are required.

Result and figure

Enter the xyz coordinates of two points in the fields on the left and press "Calculate". The result will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
- Distance \(d\) between two points in 3D space (the 3D distance formula)
- Why this formula gives the distance (using the Pythagorean theorem twice)
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the xyz coordinates of two points in space (3D), and you get the distance \(d\) between them on the spot
Along with the distance, it also finds the change in x \(\Delta x\), the change in y \(\Delta y\) and the change in z \(\Delta z\)
The result is also shown in a 3D figure you can rotate with the mouse, so you can see that the distance is the length of the line segment joining the two points (the diagonal of a box)
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

This page finds the straight-line distance between two points in space (three dimensions). For the distance in the plane, with only two coordinates (x, y), use the related "Distance Between Two Points Calculator". In 3D, a line has no single "slope" and no equation of the form y = mx + b, so this page shows only the distance and the changes in the coordinates.

What is this calculation used for?

Collision and distance checks in 3D games and VR (programming)

In a 3D game, the positions of characters, bullets and items are kept as (x, y, z) coordinates, and the game decides "it is a hit when the distance is less than a set amount" or "an enemy notices you when you come within a set distance". For example, if your character is at (1, 1, 1) and an enemy is at (2, 3, 3), the distance is \(\sqrt{1^2 + 2^2 + 2^2} = 3\). Repeating this calculation every frame is how distance checks in 3D games work.
It is just the distance formula from 2D games with one more term for \(z\).

Straight-line distance to a drone (surveying and aerial photography)

A drone's position can be given by three numbers: how far away it is horizontally (x and y) and its height (z). If a drone is 240 ft east and 320 ft north of the pilot, at a height of 300 ft, the straight-line distance to it is \(\sqrt{240^2 + 320^2 + 300^2} = \sqrt{250000} = 500\) ft.
This distance in space is what you use to think about radio range and how far away you can still see the drone.

Measuring diagonal lengths in 3D CAD and 3D printing (design and manufacturing)

In 3D CAD, positions such as the centers of holes and the corners of a part are given as coordinates in space. The distance between two points that are diagonally apart in space, which is often not written on the drawing, can be found from the differences in coordinates with the formula on this page. For example, two points 3 in apart across, 4 in apart in depth and 12 in apart vertically are \(\sqrt{3^2 + 4^2 + 12^2} = 13\) in apart.
It is an everyday calculation in making things, such as estimating the length of a brace or the shortest route for pipes and wires.

Finding the distance between atoms in a molecule (chemistry and drug discovery)

In chemistry, the position of each atom in a molecule is recorded as coordinates in space (protein databases use this format too). Once you know the coordinates of two atoms, the distance between them (a bond length, or how close two atoms are) is found with exactly the formula on this page.
It is a basic calculation behind drug discovery and materials development, for example when checking whether a candidate drug molecule fits well into its target.

Formulas and figures

Distance \(d\) between two points in 3D space (the 3D distance formula)

Figure

Standard notation (the usual math form)

\(d\) \(=\) \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\)

In words (symbols replaced with words)

② \(d\): distance between the points \(=\) ① square root of the sum of the squared differences

The formula in words

① Take the square root of the sum of the squared differences, \((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2\) (square the differences of the x-, y- and z-coordinates, add them all, and take the square root)

② and you get the \(d\): distance between the points

Quick example

The distance between the points (1, 1, 1) and (2, 3, 3) is

distance \(d\) \(=\) square root of the sum of the squared differences ((2−1)² + (3−1)² + (3−1)² = 9)

\(d = \sqrt{(2-1)^2 + (3-1)^2 + (3-1)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\)

Key idea

It is the distance formula for the plane (2D) with just one more term, \((z_2 - z_1)^2\), and the steps are exactly the same: square the differences, add them all, and take the square root only once, at the end. A common mistake is to take the square roots separately, as in \(\sqrt{(x_2-x_1)^2} + \sqrt{(y_2-y_1)^2} + \sqrt{(z_2-z_1)^2}\). Also, even if a difference of coordinates is negative, its square is positive, so the distance is the same whichever point you enter first. The answer is often a number that does not come out even, such as \(\sqrt{3}\); in that case, this calculator shows a decimal (an approximate value).

Why this formula gives the distance (using the Pythagorean theorem twice)

Figure

Standard notation (the usual math form)

\(\Delta x^2\) \(+\) \(\Delta y^2\) \(+\) \(\Delta z^2\) \(=\) \(d^2\)

In words (symbols replaced with words)

① square of the change in x, \(\Delta x\) \(+\) ② square of the change in y, \(\Delta y\) \(+\) ③ square of the change in z, \(\Delta z\) \(=\) ④ square of the distance \(d\)

The formula in words

① Add the square of the change in x, \(\Delta x\) (how far apart the points are left to right)

② the square of the change in y, \(\Delta y\) (how far apart they are front to back)

③ and the square of the change in z, \(\Delta z\) (how far apart they are up and down)

④ and you get exactly the square of the distance \(d\) between the points (the diagonal of the box) (the 3D version of the Pythagorean theorem)

Quick example

Checking with the points (0, 0, 0) and (3, 4, 12) (Δx = 3, Δy = 4, Δz = 12)

Δx squared (3² = 9) \(+\) Δy squared (4² = 16) \(+\) Δz squared (12² = 144) \(=\) distance squared (169 = 13²)

\(\ell = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)

\(d = \sqrt{\ell^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\)

Key idea

Picture a rectangular box with the two points at opposite corners. First, look only at the floor: by the Pythagorean theorem in the plane, the floor diagonal \(\ell\) satisfies \(\ell^2 = \Delta x^2 + \Delta y^2\). Next, use the Pythagorean theorem once more, on the right triangle formed by this floor diagonal \(\ell\) and the height \(\Delta z\). This gives \(d^2 = \ell^2 + \Delta z^2 = \Delta x^2 + \Delta y^2 + \Delta z^2\). So the 3D distance formula is "the Pythagorean theorem used twice", and the distance between the two points is exactly the length of the box's diagonal. Once you see that it is the same calculation as the "diagonal of a box" problem from Grade 8, you have one less thing to memorize.

To find the distance between two points in 3D space, square the differences of the x-, y- and z-coordinates, add them all, and take the square root. This is the Pythagorean theorem used twice, the same as finding the diagonal of a box. Picture it as the plane distance formula with one more term, \(\Delta z^2\), and you will not forget it.

Symbols and terms

Symbols

\(d\)	dee	The distance between the two points in space, from the first letter of "distance".
\((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\)	x sub one, y sub one, z sub one	The coordinates of the two points in space. \(x\) and \(y\) give the position on a flat plane, and \(z\) gives the height. The small number at the lower right (the subscript) tells the first point from the second.
\(\Delta x\), \(\Delta y\), \(\Delta z\)	delta x, delta y, delta z	The change (the difference of the coordinates). \(\Delta\) (delta) is a Greek letter that stands for "change" or "difference": \(\Delta x = x_2 - x_1\), \(\Delta y = y_2 - y_1\) and \(\Delta z = z_2 - z_1\).
\(\ell\)	ell	The symbol used in this explanation for the length of the floor diagonal of the box. \(\ell^2 = \Delta x^2 + \Delta y^2\); it is the in-between step when using the Pythagorean theorem twice.
\(\sqrt{\phantom{a}}\)	square root (radical sign)	The sign for the square root (the number that gives this number when squared). \(\sqrt{9} = 3\) (because 3² = 9). In the 3D distance formula, too, you take the square root only once, at the end.

Terms

3D coordinate space	Space with a z-axis for height added to the x-axis and the y-axis, where the position of a point is given by three numbers (x, y, z). It is plane (2D) coordinates with one more coordinate for height.
coordinates	Numbers that give the position of a point. In space there are three, as in the point (2, 3, 3): 2 in the x-direction, 3 in the y-direction and 3 in the z-direction (height).
line segment	A straight line that joins two points, with those points as its ends. The distance between two points is the length of this line segment.
Pythagorean theorem	The theorem that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The 3D distance formula uses this theorem twice in a row. It is taught in Grade 8.
diagonal of a box	The line that joins opposite corners of a rectangular box (rectangular prism), passing through the inside. For a box with length a, width b and height c, the diagonal is \(\sqrt{a^2 + b^2 + c^2}\) long. The distance between two points in space is the same calculation.
Euclidean distance	The formal name for the straight-line distance between two points that this page calculates. Not only in 3D but also in data analysis and AI, this name is often used for the basic way to measure how close points are.
dimension	How many numbers you need to give a position. A line is 1-dimensional (the number line), a plane is 2-dimensional (x, y), and space is 3-dimensional (x, y, z). Each extra dimension adds just one more term under the square root; the shape of the distance formula stays the same.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the fastest way forward.

The coordinate plane (Grade 6)	Being able to read coordinates such as (4, 5) as a pair made of a horizontal position and a vertical position Knowing that negative x- and y-coordinates stand for positions in the opposite direction
Operations with positive and negative numbers (Grade 7)	Being able to subtract with negative numbers, as in \(3 - (-3) = 6\) Knowing that the square of a negative number is positive
Square roots (Grade 8)	Knowing that a square root is the number that gives the original number when squared, as in \(\sqrt{9} = 3\) Knowing that a square root that does not come out even, such as \(\sqrt{3}\), can be written as an approximate decimal
The Pythagorean theorem (Grade 8)	Knowing that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse Being able to find the length of the diagonal of a box by using the Pythagorean theorem twice (this is what the 3D distance formula really is)
Solid figures and 3D coordinates (middle school and high school)	Being able to picture how the corners, edges and faces of a box are arranged Knowing that a point in space can be given by three numbers \((x, y, z)\) (this is covered in detail in high school, but the calculation on this page needs only middle school math)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the distance d between two points in 3D

Point 1 x-coordinate x₁	1
Point 1 y-coordinate y₁	1
Point 1 z-coordinate z₁	1
Point 2 x-coordinate x₂	2
Point 2 y-coordinate y₂	3
Point 2 z-coordinate z₂	3
Distance between the points d	=SQRT((B4-B1)^2+(B5-B2)^2+(B6-B3)^2)

Table to check with the Pythagorean theorem twice

Point 1 x-coordinate x₁	0
Point 1 y-coordinate y₁	0
Point 1 z-coordinate z₁	0
Point 2 x-coordinate x₂	3
Point 2 y-coordinate y₂	4
Point 2 z-coordinate z₂	12
Change in x Δx	=B4-B1
Change in y Δy	=B5-B2
Change in z Δz	=B6-B3
Floor diagonal ℓ=√(Δx²+Δy²)	=SQRT(B7^2+B8^2)
Distance between the points d=√(ℓ²+Δz²)	=SQRT(B10^2+B9^2)

After pasting, the upper rows are your inputs, and the formulas in the lower rows are calculated automatically.
"SQRT(…)" is the square root, and "^2" is squared.
The first table gives the distance 3. The second table starts from Δx = 3, Δy = 4 and Δz = 12, finds the floor diagonal ℓ = 5 first and then the distance d = 13, so you can also see the values along the way when the Pythagorean theorem is used twice.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the distance d between two points in 3D

Point 1 x-coordinate x₁	1
Point 1 y-coordinate y₁	1
Point 1 z-coordinate z₁	1
Point 2 x-coordinate x₂	2
Point 2 y-coordinate y₂	3
Point 2 z-coordinate z₂	3
Distance between the points d	=SQRT((B4-B1)^2+(B5-B2)^2+(B6-B3)^2)

Table to check with the Pythagorean theorem twice

Point 1 x-coordinate x₁	0
Point 1 y-coordinate y₁	0
Point 1 z-coordinate z₁	0
Point 2 x-coordinate x₂	3
Point 2 y-coordinate y₂	4
Point 2 z-coordinate z₂	12
Change in x Δx	=B4-B1
Change in y Δy	=B5-B2
Change in z Δz	=B6-B3
Floor diagonal ℓ=√(Δx²+Δy²)	=SQRT(B7^2+B8^2)
Distance between the points d=√(ℓ²+Δz²)	=SQRT(B10^2+B9^2)

The same formulas as in Excel (SQRT and ^2) work as is in Google Sheets. Copy the whole table, paste it into cell A1, and replace the input numbers with your own coordinates.

How to calculate it in Python

import math

x1, y1, z1 = 1.0, 1.0, 1.0   # coordinates of point 1
x2, y2, z2 = 2.0, 3.0, 3.0   # coordinates of point 2

delta_x = x2 - x1   # change in x
delta_y = y2 - y1   # change in y
delta_z = z2 - z1   # change in z
distance = math.sqrt(delta_x**2 + delta_y**2 + delta_z**2)  # distance between the points

print(f"Change in x Δx: {delta_x}")
print(f"Change in y Δy: {delta_y}")
print(f"Change in z Δz: {delta_z}")
print(f"Distance between the points d: {distance}")

# The same calculation can be written in one line with math.dist (Python 3.8 or later)
print(f"Check with math.dist: {math.dist((x1, y1, z1), (x2, y2, z2))}")

Runs with the standard library only. math.sqrt is the square root, and "**2" is squared. The last line, math.dist, is a function that finds the distance from the coordinates of two points in one step, so you can also check that the answers match. Change the coordinates at the top and run it.

How to write it in LaTeX and other math languages (copy and paste)

Distance \(d\) between two points in 3D space (the 3D distance formula)

d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>d</mi>
    <mo>=</mo>
    <msqrt>
      <mrow>
        <msup>
          <mrow><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>&#x2212;</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo></mrow>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <msup>
          <mrow><mo>(</mo><msub><mi>y</mi><mn>2</mn></msub><mo>&#x2212;</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo></mrow>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <msup>
          <mrow><mo>(</mo><msub><mi>z</mi><mn>2</mn></msub><mo>&#x2212;</mo><msub><mi>z</mi><mn>1</mn></msub><mo>)</mo></mrow>
          <mn>2</mn>
        </msup>
      </mrow>
    </msqrt>
  </mrow>
</math>

d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2)

Sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]

d := sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2);

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2);

d = √((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2)

Why this formula gives the distance (using the Pythagorean theorem twice)

Δx² + Δy² + Δz² = d²

\Delta x^2 + \Delta y^2 + \Delta z^2 = d^2

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msup><mrow><mi mathvariant="normal">&#x394;</mi><mi>x</mi></mrow><mn>2</mn></msup>
    <mo>+</mo>
    <msup><mrow><mi mathvariant="normal">&#x394;</mi><mi>y</mi></mrow><mn>2</mn></msup>
    <mo>+</mo>
    <msup><mrow><mi mathvariant="normal">&#x394;</mi><mi>z</mi></mrow><mn>2</mn></msup>
    <mo>=</mo>
    <msup><mi>d</mi><mn>2</mn></msup>
  </mrow>
</math>

(Delta x)^2 + (Delta y)^2 + (Delta z)^2 = d^2

dx^2 + dy^2 + dz^2 == d^2

dx^2 + dy^2 + dz^2 = d^2;

dx^2 + dy^2 + dz^2 == d^2

Δx^2 + Δy^2 + Δz^2 = d^2

How to have ChatGPT do the calculation

You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

For the two points (1, 1, 1) and (2, 3, 3) in 3D space (coordinate space), find each of the following:
1. The distance d between the two points (use the 3D distance formula d = √((x2−x1)² + (y2−y1)² + (z2−z1)²))
2. The change in x Δx, the change in y Δy and the change in z Δz

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula