Our three-dimensional distance calculator uses coordinates in space to calculate the distance between two places. You may use our distance calculator to compute the distance between two locations as a two-dimensional distance.

As in other math posts, we will share our knowledge and try to explain everything in this calculator and how to use it to you as easily as possible. Also, be sure to check out our other geometry tools.

## 3D Coordinate System – Definition

The three-dimensional coordinate system is comprised of an origin (normally denoted by \Theta) and three mutually perpendicular coordinate axes: the x-axis, y-axis, and z-axis.

In the rectangular coordinate system, we may find the point by utilizing the ordered pair (x, y), where x indicates its horizontal location and y represents its vertical position. Because we only need two directions to find a point, we name this system the two-dimensional coordinate system.

If we wish to find the point in space, we’ll need three numbers: x, y, and z. The ordered triple is (x, y, z). This is how we determined the origin of the 3D coordinate system. Instead of two axes, we now have three coordinate axes that are mutually perpendicular to each other.

In 3D coordinate systems, the x and y axes are often seen as two horizontal axes perpendicular to each other. However, in three dimensions, the z-axis becomes the only vertical axis.

## 2D Distance Formula

As the name implies, any distance formula provides the distance (the length of the line segment). The length of the line segment connecting two places, for instance, is the distance between them. We use the Pythagorean theorem to obtain the formula for distance between two points in a two-dimensional plane; which may also be extended to compute the distance between two points in a three-dimensional plane.

In coordinate geometry, there are several forms of distance formulas. The Euclidean distance formula is another name for the distance formula used to calculate the distance between two points in a two-dimensional plane. For example, consider two points in the 2D A(x1, y1) and B to obtain the formula (x2, y2). Assume ‘d’ represents the distance between A and B.

Derivation of Distance Formula: By the Pythagoras theorem,

AB^2=AC^2+BC^2

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

Taking the square root on both sides of the equation,

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

The formula for calculating the distance between two points is termed the distance between two points formula.

## Three Dimensional Distance – How to Find?

Consider two points in a three-dimensional plane, A (x1, y1, z1), and B(x2, y2, z2), to calculate the distance formula for two points in a three-dimensional plane. Let’s call ‘d’ the distance between points A and B.

We calculate the distance between two points in 3D using the same reasoning (as stated in the preceding section) as the distance between two points in 2D;

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

## Three-Dimensional Distance – How to Use?

Our calculator is very easy to use, and you will get the desired result very quickly. All you have to do is enter two sets of coordinates in the three-dimensional Cartesian coordinate system:(x_1, y_1, z_1) and (x_2, y_2, z_2) to get a calculation via the 2-point distance formula and calculate the distance between 2 dots.

## Three-Dimensional Distance Calculator – Example

To use our tool, you need to follow these steps:

1. Choose the coordinates of the first point – for example (1,2,3).
2. Choose the coordinates of the second point – for example, (4,6,8).
3. Input these values in the calculator, and get your results.
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

d= \sqrt{(4-1)^2+(6-2)^2+(8-3)^2} = 7,071 \; m

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## FAQ

How to find the 3D distance between two points?

Square root of sum of the squares of differences between corresponding coordinates is the distance between two locations in xyz-space, according to the distance formula. Given P1 = (x1,y1,z1) and P2 = (x2, y2, z2), so the distance d2 (P1,P2) = (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2 is the squared distance between P1 and P2. Calculate the root using this square root tool.

What is the 3D distance formula?

We calculate the distance between two points in 3D using the same reasoning (as stated in the preceding section) as the distance between two points in 2D,
d=\sqrt{[(x_2-x_1)^2+(y_2-Y_1)^2+(z_2-z_1)^2]}

How to find the distance from a point to a line in 3D?

We can find the distance from point a point to a line in the space formula: d = |M_0M_1\cdot s|

Find 3D distance between two parallel lines?

The formula for finding the distance between two parallel lines with slope-intercept form: y=mx+c_1 and y=mx+c_2, is d=\frac{|c_2-c_1|}{\sqrt{1+m^2}}