- Calculators
- ::
- Analytic Geometry
- ::
- Distance calculator

**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.

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..

**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**:

**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

Search our database with more than 250 calculators

439 092 948 solved problems