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Distance Calculator

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This calculator computes the distance between two points in two or three dimensions. It also finds the distance between two places on the world map, which are determined by their longitude and latitude. The calculator shows formulas and all steps.

Distance on a plane
find the distance between two points in plane using formula
$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
help ↓↓ examples ↓↓
$ v_1 = \left( 1, - 3 \right) ~~ v_2 = \left( -4, 2 \right) $
$ v_1 = \left( \sqrt{2},-\frac{1}{3}\right) ~~ v_2 = \left( \sqrt{5}, 0 \right) $
Distance in 3D
find the distance between two points in space using formula
$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
help ↓↓ examples ↓↓
$ v_1 = \left( 1, - 3, \sqrt{2} \right) ~~ v_2 = \left( \frac{2}{3}, -5, 2 \right) $
Longitude - latitude distance
input two point on earth surface eather in decimal or degree-minute-second form.
help ↓↓ examples ↓↓
working...
examples
example 1:ex 1:
Find the distance between the points $(–5, -1)$ and $(3, 4)$.
example 1:ex 1:
Find the distance between the points $A = \left(\dfrac{1}{2}, \sqrt{2}, 0\right)$ and $B = \left(-1, \sqrt{3}, -\sqrt{3}\right)$.
Find more worked-out examples in the database of solved problems..

About this calculator

Definition:

The distance between two points in the coordinate plane or space is the line segment length that connects these two points.


Distance in the Coordinate Plane

To find the distance between points A (X1, y1) and B (x2, y2) in a plane, we usually use the Distance formula:

$$ d(A,B) = \sqrt{(x_B - x_A)^2 + (y_B-y_A)^2} $$

Example:

Find distance between points A(3, -4) and B(-1, 3)

Solution:

First we need to identify constant x1, y1, x2 and y2: x1 = 3, y1 = -4, x2 = -1 y2 = 3. Now we can apply above formula:

d(A,B) = √[(x2 - x1)^2 + (y2-y1)^2]
d(A,B) = √[(-1-3)^2 + (3-(-4))^2
d(A,B) = √[(-4)^2 + 7^2]
d(A,B) = √[16+49]
d(A,B) = √65


Distance in the Euclidean Space

The distance between points A (X1, y1, z1) and B (x2, y2, z2) in spcace is given by the formula:

$$ d(A,B) = \sqrt{(x_B - x_A)^2 + (y_B-y_A)^2 + (z_B-z_A)^2} $$

Example:

Find distance between A(2, -1, 5) and B(3, 5, 2)

Solution:

In this example the constants are x1 = 2, y1 = -1, z1 = 5, x2 = 3, y2 = 5, z2 = 2. Now we can apply above formula:

d(A,B) = √[(x2 - x1)^2 + (y2-y1)^2 + (z2-z1)^2]
d(A,B) = √[(3-2)^2 + (5-(-1))^2 + (2-5)^2
d(A,B) = √[(1)^2 + 6^2 + (-3)^2]
d(A,B) = √[1 + 36 + 9]
d(A,B) = √46


Distance between points on sphere

The distance formula for two points on the Earth’s surface is:

d(A,B) = arccos[ sin(lat1) * sin(lat2) +cos(lat1) * cos(lat2) * cos(lon2 - lon1)] ∗6371

where lat1, lon1, lat2, and lon2 are their latitude and longitude coordinates.

Example:

Find distance between A(14.213, -38.481) and B(-2.13, 0.829)

Solution:

In this example the latitudes and longitudes: lat1 = 14.213, lon1 = -38.481, lat2 = -2.13, lon2 = 0.829. After substituting into the formula, we get:

d(A,B) = arccos[ sin(lat1) * sin(lat2) +cos(lat1) * cos(lat2) * cos(lon2 - lon1)] ∗6371
d(A,B) = arccos[ sin(14.213) * sin(-2.13) + cos(14.213) * cos(-2.13) * cos(0.829 + 38.481 )] ∗6371
d(A,B) = arccos[ - 0.2455 * 0.0371 + 0.9696 * 0.9993 * 0.7737] ∗6371
d(A,B) = arccos[ - 0.00910 + 0.7496]
d(A,B) = arccos[ 0.7405 ] ∗6371
d(A,B) = 5867.4

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