Distance Formula & Calculator

Table of contents:
If you want to know how to calculate the distance from one point to another or figure out the driving distance from A to B, then you are going to need to know the distance formula. Below, you can learn all about calculating distance in a two-dimensional space, and so much more.

Defining Distance

The definition of distance can be quite broad. The easiest way to put it is it being the one-dimensional space between one object and another. However, it can also mean more than length, in the same way that distance is not only for one dimension, but two, or even three. 

For distance to exist, you need to have two points to compare the area in between them. In mathematical terms, you would give these two points coordinates. Each coordinate is unique to that point’s position. 

Now that you have two points, you then need to figure out their accuracy and precision, as well as space with which you are measuring. The equation can differ depending on where you measure that space, but for the sake of the most common form, we will be working with Euclidean space – the default measurement option for distance.

For almost every geometric problem, we use Euclidean space, which we’ll cover in more detail below.

Euclidean Distance and the Distance Formula

When you first start learning about distances and the distance formula, we are taught about the most basic form: Euclidean distance. In this form, we know that a triangle’s angles equal 180 degrees and that a square will always have a 90-degree angle. It’s easy to assume that all measurements are like this, but those measurements don’t consider multi-dimensional spaces.

You don’t need to know why Euclidean distance is unique, and we’re not going to delve into the definition of space and the meaning of the world. However, you may like to see some examples of Euclidean space and why the other forms of distance and area may not be so relevant in our everyday lives.

Euclidean Distance and Minkowski

Before we get into how to use the distance formula calculator, it’s helpful to understand Euclidean examples next to other types of space – such as Minkowski.
Minkowski is a standard space measurement in physics. It’s similar to Euclidean but relates to relativity theory and general relativity. It differs from that due to the dot product – also called the inner product.

Both Euclidean and Minkowski deal with flat spaces, which means that the areas have flat properties – such as the distance between two straight-line points. That’s not to say that curved areas don’t exist – there are space measurements for those too! 

In a curved space measurement, the shortest distance between two spaces on a curve is not in a straight line. What’s more, instead of a flat area, you have a curved sphere, and the parallel lines on the sphere will eventually meet. 

The curved space is dramatically different from a coordinate perspective as well. Even if you take three X, Y and Z coordinates from a Euclidean space and turn it into a spherical coordinate; the Euclidean feature remains the same. However, in a curved area, the measurements are different and not as easy to express in a number form.

Hopefully, that makes some sense, but if not, this distance formula from the standard Euclidean space might help. 

√ [(x₂ - x₁)² + (y₂ - y₁)²]

The distance formula shares a close link to the Pythagorean theorem of a² + b² = c². In this case, a and b would form the right triangle’s legs, and c would be the hypotenuse. 

Consider that (x₁, y₁) and (x₂, y₂) are the two distance-creating coordinates of the hypotenuse (c). In that case, (x₂ - x₁)² would link to a² and (y₂ - y₁)² with . What’s more, as c equals √ (a² + b²), it’s not hard to make the connection that the distance formula is a Pythagorean theorem extension.

The Distance from a Continuous Structure or Straight Line

What do you do if you don’t want to find the distance from one point to another, but rather from an end to a circle or a line? You would then need to identify the first point from which to measure the distance, and then use the distance formula to calculate it. 

To do that, you need to know how a perpendicular line works, as a point and continuous object’s path is something you would define perpendicularly. The first step is to measure the distance from one location to another, join the ends with a line, then measure that part’s length. How do you know which point from which to draw though? It should be the point which is perpendicular to your first line to which you measure the distance. 

If a distance point is part of a line, the distance will be zero. If the distance you are measuring is one-dimensional, the line signifies the entire one-dimensional area, so you can only measure the distance between those points. 

Because of this restriction, there are limitations on how to measure distances. Knowing the above information, you can then end up changing the concept of how tall a triangle is. You would be able to measure the distance from a vertex to the opposite side, redefining the distance terms because of the area playing the part of the function of the triangle.

If that’s not enough to confuse you, then you also have to consider the midpoint while using the distance formula. The midpoint is the central spot between two points. It’s the same distance from each of the two points. As a rule of thumb, if you use a midpoint in a circle, it represents the center of the object.

Using a Distance Calculator to Find the Distance

Even though the distance formula calculator can take care of the entire equation for you, it’s not a silly idea to learn how to work it out by hand. Otherwise, what are you going to do if you need to figure it out without having the calculator handy? 

We will focus on two-dimensional space as previously mentioned; it’s the most commonly used in our daily lives. In a two dimensional setting, you use two coordinates – one at each end for measuring the distance between the two.

Follow these steps below.

   1.	Put your data into this distance formula 
√[(x₂ - x₁)² + (y₂ - y₁)²]
   2.	In parentheses, subtract values 

   3.	Square both quantities

   4.	Add the results together

   5.	Take the square root

   6.	Check your results with the distance formula calculator

A Road Trip Example Use of a Distance Formula

You may not think you’ll end up using a distance formula or calculator all that much, but it can end up being quite a useful tool – especially if you pair it with a gas calculator for travelling. 

Let’s say you’re driving to your mom’s for Thanks giving from your house, and you’re making one stop to pick up your sister. You need to travel from City A to City B, with a side route of City C. The roads to get to these destinations are not going to be perfectly straight, which can make finding out your distance quite tricky. 

You won’t be able to get an entirely accurate depiction of the distance because a straight line is only going to be approximate. In this case, Google Maps or a path-measuring tool like it can be far more effective than using distance formulas. 

You would only use a distance formula calculator for proper measurements – such as those in a straight line between two objects, rather than busy winding roads. You would, however, use a distance formula for measuring the distance of the Earth to the Sun and Moon.

Astronomical Distances: The Distance from the Earth to the Sun and Moon

The distance between Earth and other planets has been of great interest to people for thousands of years. While it’s hard to measure distances for what’s around us, it tends to be straightforward when you measure the distance between something off in the distance.

To measure the distance from the Earth to the sun or moon, you use the standard Euclidean distance, coupled with light years as a measurement and scientific notation. That’s how people have already worked out that Earth to moon distance is over 238,000 miles, and the sun to the Earth distance is around 932,056,78 miles or eight light years. 

When you start getting into higher measurement figures, you’ll see the entire universe is about 46,600,000,000 light years in size. At that point, it’s helpful to begin measuring distance in time as opposed to length.

Distance Beyond the Length Measurement

When you talk about Euclidean distance, you know that the distance refers to length. However, when you start talking about light years and differences between one point and another, an entirely new realm opens up. You no longer have to think of everything as a measurement of length, but rather what lies between one point and another.

For example, consider the one-dimensional difference between numbers three and four. There is no space between them, but there is something. To know what that is, you would take one away from the other to get the difference (or distance). 

You could say the same about a percentage difference, which you can use a calculator for too. It’s another way of gathering the distance or gap between two things. 

If, however, you were to take away the concept of distance entirely and replace it with a difference, you could get into temperatures, pressure, and even thermal energy. However, you can also measure distance by way of time and the method of transport to get you from point A to B. 

If you traveled 100 miles by car, it would not take the same time as it would by train, bicycle, or plane. But you would still go the same distance. Not to get too far away from the original idea of distance formulas, but it’s clear to see the possibilities are endless for measuring distance beyond the restraints of length.

A distance formula is something that can end up being a valuable tool in your daily life. Once you grasp the concept of distance measurements, the standard length measurement ends up being only the starting point.

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