Formula and Execution

(x1

y1) and (x2, y2),These are the coordinates of the points you enter to calculate the distance between two points.

Distance

This section provides the required results of the distance. The formula that is set up to work for this calculator is the Euclidean distance formula: d = √[(x2 – x1)2+(y2 – y1)2].

Distance Calculator

Well, we all need to know the distance whenever we are traveling from one place to another. Knowing the distance makes it easy for us to evaluate the time that would be required to reach our destination. With the advancement of technology, we have many mobile phone applications and tools these days that work for us to make our calculations hassle-free for us. The distance is just limited to this, but we have many more applications that will be discussed later in the article.

There always exists the theory behind these modern-day hassle-free calculation tools. Somewhere, someone worked and proposed some theory based on observations that we are usually taught at schools. 

And when these school ventures ask you to calculate the distance between two points in 2D space, never hesitate to visit our online distance calculator for no wait results. It does not even take a fraction of second to compute the distance between two points after the values are added to it.

How to Calculate the Distance Between Two Points Using Calculator Beast

Our distance calculator works for points in a 2D Cartesian plane. The formula for distance is d = √[(x2 – x1)2+(y2 – y1)2]. The calculator at your left-hand side already shows you the columns for x1, x2, y1, and y2. Simply substitute the coordinates of the points in their respective place, and there you get quick results for distance. 

A screenshot of the calculator is attached for a better understanding of how it works. The points selected are: (-2, 0) and (1, 4) and the result is 5 units.

Distance Calculator

Concept of Distance

We discussed how our calculator works to calculate the distance between two points. From this section onwards, we will dive a bit deep into the ocean of concepts of distance

Defining Distance

When it comes to defining distance, no rocket-science is needed to do that. It is a very basic definition that even kids would be easily able to interpret. Distance is simply the measure to calculate how far two objects are located. If we are to calculate the distance between two points A and B, then the distance from A to B is interchangeable with distance from B to A. This numeric quantity tells us how far or how near a thing is.

Example:

Consider an object moved from position 2 to position 7, as shown in the figure.

 

Defining Distance

 

To calculate the distance from 2 to 7, we simply subtract 2 from 7, i.e., 7-2=5, so, 5 units is the distance when the object is moved from 2 to 7.

What if the object is moved from 7 to 2? Well, the distance is still the same. Although we would subtract 7 from 2, i.e., 2-7=-5, remember the distance is always positive, and to make it positive, we apply absolute value operation on it.

Absolute Value

Absolute value is the operation in math that changes the negative sign to positive. When absolute value is applied to zero or positive, there is no change. The absolute value is symbolized by vertical brackets like |-5| = 5. Positive values are also symbolized in a similar fashion, i.e., |5| = 5.

So, since the distance between anything can never be negative, so it is the absolute value of the result.

Directed Distance

Directed distance is the same as the distance; the only difference is that it can be negative. Well, in the distance, we talked that it can never be negative, so why is directed distance negative? Because it shows the direction too.

Like in the example shown in the distance definition, we moved an object from 2 to 7 and then 7 to 2. The distance in both cases was 5, while the directed distance is 5 and -5, respectively, where the positive sign shows movement from 2 to 7, while the negative sign shows movement from 7 to 2.

Directed distance is measured along straight lines as well as curved paths

Displacement

Displacement is a directed distance considered along a straight line.

Psychological Distance

Psychological distance is not a numeric quantity. Psychological distancing means to remove a thing, person, or an event from personal dimensions such as time and space.

Well, we won’t be discussing the psychological distances here, because we are just concerned with the mathematical definitions of distances.

Kinds of Physical Distances

Distance Travelled

Distance traveled is simply the length of the path you travel while going from one place to another.

Euclidean Distance

Euclidean distance is simply the straight-line distance between two points in a Euclidean space. The Euclidean formula for distance in the 2D plane where the coordinates of the two points are (x1, y1) and (x2, y2), is,

d = √[(x2 – x1)2+(y2 – y1)2]

If we are to calculate the distance in Euclidean n-space, and we consider the points to be (x1, x2,…,xn) and (y1, y2,…,yn), the formula follows the same pattern and can be written as:

d = √[(x1 – y1)2+(x2 – y2)2+…(xn – yn)2]

Manhattan Distance

If we take the absolute difference between the Cartesian coordinates of two points and sum them up together, we get Manhattan distance. The Figure below shows the Manhattan distance and the Euclidean distance between two points that are dotted black. The red, blue, and yellow lines show the Manhattan distance, while the green line shows the Euclidean distance. 

Manhattan Distance

For the red line, we sum up two absolute differences between Cartesian coordinates; for the yellow line, we sum up four absolute differences between Cartesian coordinates, and similarly, for blue distance, we sum up twelve absolute differences. 

The name Manhattan distance was given to it because the streets of New York City are divided into grids, and anyone traveling from one place to another has to travel on the grid of streets to reach their destination.

Aerial Distance

Aerial distance is the distance between two points on the ground, but it is the distance that is measured in the air. These distances are measured by the path that airplanes travel from one place to another.

Aerial distances are always shorter than the distances on the grounds, that is why it takes 3 hours and 50 minutes to travel from NYC to WDC by car and 1 hour 10 minutes to travel by air.

Circular Distance

The circular distance is the distance traveled by a wheel. This distance helps to design vehicles and mechanical gears. This is also the distance that is traveled by the Earth when it completes one rotation and the distance traveled by the ball when it is thrown in the air, and it returns to the initial position. 

Geodesic Distance

It is the shortest distance between two points on a curved surface. The distances measured on the surface of the Earth is an example. 

Chessboard Distance

This distance is the minimum number of moves a king makes on a chessboard while traveling from one square to another. Interesting, right?

Distance in Cosmology

These are distances between two objects in the space and the distances between any two events that take place in the universe. These distances are further classified into many kinds.

FAQs

How is the Euclidean formula for distance derived?

To derive Euclidean formula for distance, consider two points A(x1, y1) and B(x2, y2) in the Cartesian coordinates system. To find the length of the AB, i.e., the distance of this line, we draw a right-angled triangle where AB is the hypotenuse. To draw the triangle, we drop a perpendicular from point B to point C on the x-axis and then draw another perpendicular on BC from point A as shown in the figure.

How is the Euclidean formula for distance derived

Now, as we see, triangle ABD is a right triangle with coordinates A(x1, y1), B(x2, y2), and D(x2, y1). To find the length of the hypotenuse AB, we use the Pythagoras theorem. 

  • AB2 = AD2 + BD2
  • AB = √[AD2 + BD2]

The length of AD is x2-x1, and the length of BD is y2-y1. Substituting these values in the above equation, we get

  • AB = √[(x2-x1)2 + (y2-y1)2]

The above is the distance formula for Euclidean distance.

Can distance be negative?

No, distance can never be negative. Distance is a scalar quantity that gives us the magnitude of how far an object is from another object, or how far an object has moved from its original position.

Directed distance or displacement, on the other and, can be negative. It is a vector quantity that not only gives us the magnitude of the distance between two things, but it also shows us the direction of the movement of the object. If the object moves in the right direction, it will be positive displacement, and if it moves towards the left direction, it will be negative. 

Distance, in other words, is the absolute value of directed distance.

What are the real-life applications of distance formula?

Distance formula has many applications that can be seen in the real world. Some of them are listed below.

  1. The distance formula is useful when finding the distances between two towns, cities, or countries on a map. Just calculating the coordinates of each place and substituting them in the distance formula yields the results for the distance between them.
  2. After computing all the coordinates, we can calculate the distance between the home base and the second base in a baseball field.
  3. The distances of all the throws in sports are calculated using the distance formula.
  4. If the height of a lighthouse and the distance between the ship and the base of the lighthouse are known, then the distance between the observer at lighthouse and ship can be calculated using the distance formula.

Fun Facts About Distances

  1. If all the DNA in your cells were laid out in a straight line, it would reach to the sun and back around 70 times.
  2. The distance between the Earth and the sun is 146 million km.
  3. The distance between the Earth and the moon is 384,400 km.
  4. The circumference of the Earth around equator is 40,030 km.
  5. Your body contains around 96,000 km of blood vessels.
  6. The giant jellyfish can make its tentacles reach around 120 feet.
  7. The longest street in the world, Yonge Street, is 1896 km long. It is in Toronto, Canada.
  8. The length of The Great Wall of China is around 6430 km long.
  9. The length of the small intestines of the ostrich is 46 feet.
  10. In his entire lifetime, an average human walks as much as he would need to walk around the world twice.