# Distance Calculator

## How to use a 2D Distance Calculator?

The 2D Calculator calculates the Distance between two points of a 2D coordinate plane.

In Point 1, add the value of x1 and y1.

In Point 2, add the value of x2 and y2. #### How to use a 3D Distance Calculator?

It is similar to the 3D Distance Calculator. However, the 3D Calculator helps you determine the Distance between two points of a 3D coordinate space.

In Point 1, add the value of x1, y1, and z1.

In Point 2, add the value of x2, y2, and z2.

Distance is determined based on longitude and latitude.

Using the Decimal Calculator helps to find the shortest distance between the points of an Earth's surface.

The Decimal is based on Degree, Minute, and Second.

From Point 1, add the First Latitude and Longitude Values.

In Point 2, add the Second Latitude and Longitude Values.

#### 1. Based on a 2D Coordinate Plane

So the Distance between the two points of a 2D coordinate plane can be calculated using the following formula/equation.

d = √(x2 - x1)2 + (y2 - y1)2

Now (x1, y1) and (x2, y2) -coordinates of the two respective points.
The order of the point is not important in the formula. However, the points have to be consistent.

So let's consider, for example.

Two points (1, 5) and (3, 2)

So, any number 1 or 3 could be considered as x1 or x2 as long as the corresponding y values are used.

Let's use (1, 5) as (x1, y1) and (3, 2) as (x2, y2).

In the formula d = √(x2 - x1)2 + (y2 - y1)2

d = √(3 - 1)2 + (2 - 5)2

= √22 + (-3)2

= √4 + 9

= √13

Similarly, Let's use (3, 2) as (x1, y1) and (1, 5) as (x2, y2)
Using the same formula

d = √(x2 - x1)2 + (y2 - y1)2

d = √(1 - 3)2 + (5 - 2)2

= √(-2)2 + 32

= √4 + 9

= √13

So in both cases, the answer or the result is the same.

#### Let's calculate Distance in a 3D Coordinate Space

The Distance between two points of a 3D coordinate space can be determined using the following formula.

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

Here (x1, y1, z1) and (x2, y2, z2) are the 3D coordinate spaces of the two points.

Similar to the 2D formula, it doesn't matter where the two points are indicated (x1, y1, z1) or (x2, y2, z2).

So add the values in the formula and calculate.

The calculation is as follows:

d = √(2 - 1)2 + (4 - 3)2 + (8 - 7)2

= √12 + 12 + 12

= √3

#### Calculating distance from one end to another on the Earth’s Surface

There are many ways to calculate the Distance on an Earth's Surface. But the two Haversine and Lambert Formulas are widely used.

The Haversine formula determines the Distance between two points of a sphere. The longitude and latitude should be Haversine Formula.

d = 2rsin-1 sin² (21)+ 71) ) + cos(41) cos(2) sin² (^2^1)

In the formula, d is the Distance between two points on a big circle.

ϕ1and ϕ2 are the latitudes of the two points.

λ1 and λ2 are the longitudes of those two points.

The Haversine Formula helps determine the great Distance between the points of longitude and latitude of a sphere.

It helps in determining the approximate distance of the Earth.

It is the largest sphere that can be anywhere on the given sphere.

It is formulated by the intersecting planes and sphere with the sphere's center.

So the Great circle is the shortest Distance between two points of a sphere.

The answers or results using the Haversine Formula can have some error upto 0.5% as the Earth is a spherical or perfect shape. It is an ellipsoid with a radius of 6,378 km at the Equator. It has a radius of 6,357 km on the pole.

Lambert's formula, an ellipsoidal-surface formula, gives a more accurate approximate surface value.

So a Lambert Formula can't be used instead of the Haversine Formula.

#### 2. Lambert's formula

The Lambert formula is used to calculate the shortest Distance on the surface of an Ellipsoid. It gives an accuracy of 10 meters.
Lambert's formula is

dot d = a(sigma - f/2 * (X + Y))

Here is the equatorial radius of the ellipsoid.

But here it is the Earth

σ=central angle between the radians of longitude and latitude.

f=flattening of the Earth.

And X and Y can be expanded in the following way.

X = (sigma - sin(sigma)) * (sin^2 (P) * cos^2 (Q))/(cos^2 (sigma/2))

Y = (sigma + sin(sigma)) * (cos^2 (P) * sin^2 (Q))/(sin^2 (sigma/2))

Here P=(β1 + β2)/2 and Q= β2 + β1/2.

So, in the equation, β1 and β2 are the minimized latitudes after using the formula.

tan(β) = (1 - f)tan(ϕ).

As here, ϕ is the latitude of a given point.

#### To Conclude

Here the formulas of Haversine and Lambert don’t provide accurate and exact details that are impossible to account for the irregularity of the Earth's surface. But these equations and formulas can help determine and approximate close to the Earth.

The Distance Calculator on AllCalculator.net offers accurate and efficient calculations for determining distances between points in various dimensions, including 2D, 3D, and on the Earth's surface.

### Why is a 2D Distance Calculator used?

A 2D Distance Calculator helps to determine the distance between two points. It shows the shortest distance on a map.

### What is a 3D Distance Calculator?

The 3D Distance Calculator calculates the distance between two points in a 3D coordinate space. d√(x2-x1)2-(y2-y1)2-(z2-z1)2 is used to calculate the 3D coordinate Distance.

### How is the Distance of Earth's Surface based on latitude and longitude?

The distance on the Earth's surface is usually based on longitude and latitude. So to calculate the shortest distance between two points, the latitude and longitude of an earth's surface, the calculator is used.

### How does a Distance Calculator work?

A Distance Calculator can calculate the distance between any two points, like the distance between two cities or locations.

### What is the Lambert formula?

It calculates the short distance on the Surface of a circle or ellipsoid. The formula is d = a(σ - f/2 * (X + Y))

### What is a haversine formula? Why is it used?

A haversine formula calculates the distance between two points on the Earth's Surface.

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