Distance Calculator & Formula

Have you ever wanted khổng lồ calculate the distance from one point to lớn another, or the distance between cities? Have you ever wondered what the distance definition is? We have all these answers và more, including a detailed explanation of how to lớn calculate the distance between any two objects in 2d space. As a bonus, we have a fascinating topic on how we perceive distances (for example as a percentage difference); we"re sure you"ll love it!

Prefer watching over reading? Learn all you need in 90 seconds with this video clip we made for you:


What is distance?


Before we get into how khổng lồ calculate distances, we should probably clarify what a distance is. The most common meaning is the /1D space between two points. This definition is one way khổng lồ say what almost all of us think of distance intuitively, but it is not the only way we could talk about distance. You will see in the following sections how the concept of distance can be extended beyond length, in more than one sense that is the breakthrough behind Einstein"s theory of relativity.

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If we stick with the geometrical definition of distance we still have lớn define what kind of space we are working in. In most cases, you"re probably talking about three dimensions or less, since that"s all we can imagine without our brains exploding. For this calculator, we focus only on the 2 chiều distance (with the 1D included as a special case). If you are looking for the 3 chiều distance between 2 points we encourage you to use our 3 chiều distance calculator made specifically for that purpose.

To find the distance between two points, the first thing you need is two points, obviously. These points are described by their coordinates in space. For each point in 2 chiều space, we need two coordinates that are quality to that point. If you wish to lớn find the distance between two points in 1D space you can still use this calculator by simply setting one of the coordinates to lớn be the same for both points. Since this is a very special case, from now on we will talk only about distance in two dimensions.

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The next step, if you want to be mathematical, accurate, and precise, is lớn define the type of space you"re working in. No, wait, don"t run away! It is easier than you think. If you don"t know what space you"re working in or if you didn"t even know there is more than one type of space, you"re most likely working in Euclidean space. Since this is the "default" space in which we do almost every geometrical operation, và it"s the one we have set for the calculator to lớn operate on. Let"s dive a bit deeper into Euclidean space, what is it, what properties does it have and why is it so important?


The Euclidean space or Euclidean geometry is what we all usually think of 2d space is before we receive any deep mathematical training in any of these aspects. In Euclidean space, the sum of the angles of a triangle equals 180º và squares have all their angles equal lớn 90º; always. This is something we all take for granted, but this is not true in all spaces. Let"s also not confuse Euclidean space with multidimensional spaces. Euclidean space can have as many dimensions as you want, as long as there is a finite number of them, and they still obey Euclidean rules.

We vì not want to bore you with mathematical definitions of what is a space và what makes the Euclidean space unique, since that would be too complicated to lớn explain in a simple distance calculator. However, we can try khổng lồ give you some examples of other spaces that are commonly used & that might help you understand why Euclidean space is not the only space. Also, you will hopefully understand why we are not going to bother calculating distances in other spaces.

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The first example we present to you is a bit obscure, but we hope you can excuse us, as we"re physicists, for starting with this very important type of space: Minkowski space. The reason we"ve selected this is because it"s very common in physics, in particular it is used in relativity theory, general relativity and even in relativistic quantum field theory. This space is very similar lớn Euclidean space, but differs from it in a very crucial feature: the addition of the dot product, also called the inner product (not to lớn be confused with the cross product).

Both the Euclidean và Minkowski space are what mathematicians điện thoại tư vấn flat space. This means that space itself has flat properties; for example, the shortest distance between any two points is always a straight line between them (check the linear interpolation calculator). There are, however, other types of mathematical spaces called curved spaces in which space is intrinsically curved và the shortest distance between two points is no a straight line.

This curved space is hard khổng lồ imagine in 3D, but for 2d we can imagine that instead of having a flat plane area, we have a 2 chiều space, for example, curved in the shape of the surface of a sphere. In this case, very strange things happen. The shortest distance from one point khổng lồ another is not a straight line, because any line in this space is curved due lớn the intrinsic curvature of the space. Another very strange feature of this space is that some parallel lines bởi actually meet at some point. You can try to lớn understand it by thinking of the so-called lines of longitude that divide the Earth into many time zones và cross each other at the poles.

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It is important to chú ý that this is conceptually VERY different from a change of coordinates. When we take the standard x, y, z coordinates & convert into polar, cylindrical, or even spherical coordinates, but we will still be in Euclidean space. When we talk about curved space we are talking about a very different space in terms of its intrinsic properties. In spherical coordinates, you can still have a straight line và distance is still measured in a straight line, even if that would be very hard to lớn express in numbers.

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Coming back khổng lồ the Euclidean space, we can now present you with the distance formula that we promised at the beginning. The distance formula is

√<(x₂ - x₁)² + (y₂ - y₁)²>,

which relates to the Pythagorean theorem: a² + b² = c². Here, a & b are legs of a right triangle and c is the hypotenuse. Suppose that two points, (x₁, y₁) and (x₂, y₂), are coordinates of the endpoints of the hypotenuse. Then (x₂ - x₁)² in the distance equation corresponds khổng lồ a² & (y₂ - y₁)²corresponds to b². Since c = √(a² + b²), you can see why this is just an extension of the Pythagorean theorem.

An extended application of the distance between points can be found in the segment addition postulate, which involves finding a segment length when 3 points are collinear.


The distance formula we have just seen is the standard Euclidean distance formula, but if you think about it, it can seem a bit limited. We often don"t want to find just the distance between two points. Sometimes we want khổng lồ calculate the distance from a point khổng lồ a line or to lớn a circle. In these cases, we first need khổng lồ define what point on this line or circumference we will use for the distance calculation, và then use the distance formula that we have seen just above.

Here is when the concept of perpendicular line becomes crucial. The distance between a point & a continuous object is defined via perpendicularity. From a geometrical point of view, the first step khổng lồ measure the distance from one point to another, is lớn create a straight line between both points, và then measure the length of that segment. When we measure the distance from a point lớn a line, the question becomes "Which of the many possible lines should I draw?". In this case the answer is: the line from the point that is perpendicular lớn the first line. This distance will be zero in the case in which the point is a part of the line. For these 1D cases, we can only consider the distance between points, since the line represents the whole 1D space.

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This imposes restrictions on how khổng lồ compute distances in some interesting geometrical instances. For example, we could redefine the concept of height of a triangle khổng lồ be simply the distance from one vertex khổng lồ the opposing side of the triangle. In this case, the triangle area gets also redefined in terms of distance, since the area is a function of the height of the triangle.


Let"s look at couple examples in 2 chiều space. Khổng lồ calculate the distance between a point and a straight line we could go step by step (calculate the segment perpendicular lớn the line from the line to lớn the point và the compute its length) or we could simply use this "handy-dandy" equation: d = |Ax1 + By1 + C | / √(A2 + B2) where the line is given by Ax+By+C = 0 và the point is defined by (x1, y1).

The only problem here is that a straight line is generally given as y = mx + b so we would need to lớn convert this equation lớn the previously show form: y = mx + b → mx - y + b = 0 so we can see that A = m, B = -1 and C = b. This leaves the previous equation with the following values: d = |mx1 -y1 + b | / √(m2 + 1).

For the distance between 2 lines, we just need to lớn compute the length of the segment that goes from one to the other và is perpendicular to lớn both. Once again, there is a simple formula to help us: d= |C2-C1|/√(A2+B2) if the lines are A1 x+B1 y+C1 = 0 and A2x+B2y+C2 = 0. We can also convert khổng lồ slope intercept for and obtain: d= |b2-b1|/√(m2+1) for lines y = m1x + b1 và y = m2x + b2.

Notice that both line needs to be parallel since otherwise the would cảm ứng at some point và their distance would then be d = 0. That"s the reason the formulas omit most of the subscripts since for parallel lines: A1 = A2 = A & B1 = B2 = B while in slope intercept khung parallel lines are those for which m1 = mét vuông = m

Apart from perpendicularity, another important concept to lớn talk about regarding distance, is the midpoint. This is the point that is precisely in the middle between the two others. The midpoint is defined as the point that is the same distance away from each of the points of reference. We can and will generalize this concept in a later section, but for now, we can limit ourselves lớn geometry. For example, the midpoint of any diameter in a circle or even a sphere is always the centre of said object.


As we have mentioned before, distance can mean many things, which is why we have provided a few different options for you in this calculator. You can calculate the distance between a point và a straight line, the distance between two straight lines (they always have to be parallel), or the distance between points in space. When it comes khổng lồ calculating the distances between two point, you have the option of doing so in 1, 2, 3, or 4 dimensions. I know, I know, 4 dimensions sounds scary, but you don"t need to lớn use that option. & you can always learn more about it by reading some nice resources và playing around with the calculator. We promise it won"t break the mạng internet or the universe.

We have also added the possibility for you khổng lồ define 3 different points in space, from which you will obtain the 3 pairs of distances between them, so, if you have more than two points, this will save you time. The number of dimensions you are working in will determine the number of coordinates that describe a point, which is why, as you increase the number of dimensions, the calculator will ask for more input đầu vào values.

Even though using the calculator is very straightforward, we still decided lớn include a step-by-step solution. This way you can get acquainted with the distance formula and how khổng lồ use it (as if this was the 1950"s và the internet was still not a thing). Now let"s take a look at a practical example: How to lớn find the distance between two points in 2-D.

Suppose you have two coordinates, (3, 5) & (9, 15), & you want to lớn calculate the distance between them. Khổng lồ calculate the 2-D distance between these two points, follow these steps:

Input the values into the formula: √<(x₂ - x₁)² + (y₂ - y₁)²>.Subtract the values in the parentheses.Square both quantities in the parentheses.Add the results.Use the distance calculator to kiểm tra your results.

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Working out the example by hand, you get:

√<(9 - 3)² + (15 - 5)²> = √<(6)² + (10)²> = √<36 + 100> = √136,

which is equal lớn approximately 11.66. Note, that when you take the square root, you will get a positive and negative result, but since you are dealing with distance, you are only concerned with the positive result. The calculator will go through this calculations step by step lớn give you the result in exact và approximate formats.


Let"s take a look of one of the applications of the distance calculator. You can use it together with the gas calculator for making road trip plans. Suppose you are traveling between cities A and B, và the only stop is in đô thị C, with a route A khổng lồ B perpendicular khổng lồ route B to lớn C. We can determine the distance from A to lớn B, & then, with the gas calculator, determine fuel cost, fuel used & cost per person while traveling.

The difficulty here is to calculate the distances between cities accurately. A straight line (like what we use in this calculator) can be a good approximation, but it can be quite off if the route you"re taking is not direct but takes some detour, maybe lớn avoid mountains or to pass by another city. In that case, just use Google maps or any other tool that calculates the distance along a path not just the distance from one point lớn another as the crow flies.

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Where our calculator can give proper measurements & predictions, is when calculating distances between objects, not the length of a path. With this in mind, there are still multiple scenarios in which you might actually be interested in the distance between objects, regardless of the path you would have lớn take. One such example is the distance between astronomical objects.


When we look at a distance within our Earth, it is hard lớn go far without bumping into some problems, from the intrinsic curvature of this space (due khổng lồ the Earth curvature being non-zero) khổng lồ the limited maximum distance between two points on the Earth. It is because of this, & also because there is a whole universe beyond our Earth, that distances in the universe are of big interest for many people. Since we have no proper means of interplanetary traveling, let alone interstellar travels, let"s focus for now on the actual Euclidean distance khổng lồ some celestial objects. For example the distance from the Earth lớn the Sun, or the distance from the Earth to the Moon.

These distances are beyond imaginable for our ape-like brains. We struggle khổng lồ comprehend the size of our planet, never mind the vast, infinite universe. This is so difficult that we need to use either scientific notation or light years, as a unit of distance for such long lengths. The longest trips you can vì chưng on Earth are barely a couple thousand kilometers, while the distance from Earth khổng lồ the Moon, the closest astronomical object to lớn us, is 384,000 km. On top of that, the distance khổng lồ our closest star, that is the distance from Earth to the Sun, is 150,000,000 km or a little over 8 light minutes.

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When you compare these distances with the distance to lớn our second nearest star (Alpha Centauri), which is 4 light years, suddenly they start lớn look much smaller. If we want to go even more ridiculous in comparison we can always think about a flight from new york to Sydney, which typically takes more than 20 h & it"s merely over 16,000 km, & compare it with the size of the observable universe, which is about 46,600,000,000 light years!

Here, we have inadvertently risen a fascinating point, which is that we measure distances not in length but in time. Thus, we extend the notion of distance beyond its geometrical sense. We will explore this possibility in the next section as we speak about the importance & usefulness of distance beyond the purely geometrical sense. This is a very interesting path to take & is mostly inspired by the philosophical need to extend every concept khổng lồ have a universal meaning, as well as from the obvious physical theory to lớn mention, when talking about permutations of the space và time, or any other variable that can be measured.


Typically, the concept of distance refers lớn the geometric Euclidean distance and is linked to lớn length. However, you can extend the definition of distance to lớn mean just the difference between two things, & then a world of possibilities opens up. Suddenly one can decide what is the best way khổng lồ measure the distance between two things và put it in terms of the most useful quantity. A very simple step to lớn take is to lớn think about the distance between two numbers, which is nothing more than the 1D difference between these numbers. To lớn obtain it, we simply subtract one from the other và the result would be the difference, a.k.a. The distance.

We could jump from this numerical distance to, for example, difference or distance in terms of the percentage difference, which in some cases might provide a better way of comparison. We don"t need to stay just with percentage, we can convert percentage khổng lồ fraction if you feel that would be the best way khổng lồ express distance. But so far, this is still just one level of abstraction in which we simply remove the units of measurement. But what if we were to lớn use different units altogether?

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By extending the concept of distance lớn mean something closer to difference, we can calculate the difference between two temperatures in terms of degrees or thermal energy, or other related quantity lượt thích pressure. But we don"t need khổng lồ get really extreme, let"s see how two points can be separated by a different distance, depending on the assumptions made. Coming back to the driving distance example, we could measure the distance of the journey in time, instead of length. In this case, we need an assumption lớn allow such translation; namely the way of transport.

There is a big difference in the time taken to travel 10 km by plane versus the time it takes by car; or by oto versus bike. Sometimes, however, the assumption is clear & implicitly agreed on, lượt thích when we measure the lightning distance in time which we then convert to lớn length. This brings up an interesting point, that the conversion factor between distances in time & length is what we hotline speed or velocity (remember they are not exactly the same thing). Truth be told, this speed doesn"t have lớn be constant as exemplified by accelerated motions such as that of a miễn phí fall under gravitational force, or the one that links stopping time và stopping distance via the breaking force and drag or, in very extreme cases, via the force of a oto crash.

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Another place where you can find weird units of distance are in solid state physics, where the distance a particle travels inside of a material is often expressed as an average of interactions or collisions. This distance is linked to lớn length by using the mean không tính phí path which is the mean distance (in length) a particle travels between interactions. If we want lớn get even more exotic we can think about the distance from the present value to the future value of something like a car. This distance between prices is linked here by the oto depreciation, và it"s not as cut và dry as the other distances, but only because of the number of factors involved in calculating this distance.

We don"t want to, however, make anyone"s brain explode, so please don"t think too hard about this. Just take this calculator & use it for length-based distance in 2d space. You can always return khổng lồ this philosophical view on distances if you ever find yourself bored và having already checked all of our Omni Calculators.


To find the distance between two points we will use the distance formula: √<(x₂ - x₁)² + (y₂ - y₁)²>

Get the coordinates of both points in spaceSubtract the x-coordinates of one point from the other, same for the y componentsSquare both results separatelySum the values you got in the previous stepFind the square root of the result above.

If you think this is too much effort you can simply use the Distance Calculator from Omni


Distance is not a vector. The distance between points is a scalar quantity, meaning it is only defined by its value. However, the displacement is a vector with value và direction. So the distance between A and B is the same as the distance from B to lớn A, but the displacement is different depending on their order.


Click is slang for a kilometer which is 0.62 miles. It is actually written with "k" (Klick) as it is derived from the word kilometer. It is commonly used in the military và motorcyclists.


The distance formula is: √<(x₂ - x₁)² + (y₂ - y₁)²>. This works for any two points in 2 chiều space with coordinates (x₁, y₁) for the first point & (x₂, y₂) for the second point. You can memorize it easily if you notice that it is Pythagoras theorem và the distance is the hypothenuse, and the lengths of the catheti are the difference between the x and y components of the points.


The distance of a vector is its magnitude. If you know its components:

Take each of the components of the vector and square themSum them upFind the square root of the previous resultEnjoy the good work!

If you know its polar representation, it will be a number và an angle. That number is the magnitude of the vector, which is its distance.


The ham mê unit of distance is the meter, abbreviated lớn "m". A meter is approximately 3.28 feet. Other common units in the International System of units are the centimeter (one one-hundredth of a meter, or 0.39 inches) & the kilometer (one thousand meters or 0.62 miles), among others.


The distance from A to B is the length of the straight line going from A khổng lồ B. The distance from B khổng lồ A is the same as the distance from A lớn B because distance is a scalar


Distance is a measure of one-dimensional space. The distance between two points is the shortest length of 1D space between them. If you divide distance over time you will get speed, which has dimensions of space over time.


A light-year is a measurement of distance. It is 9.461⨉1012 kilometers or 5.879⨉1012, which is the distance traveled by a ray of light in a perfect vacuum over the span of a year.


The velocity & the moving time of an object you can calculate the distance:

Make sure the speed and time have compatible units (miles per hour và hours, meter per second, and seconds...)If they aren"t, convert them lớn the necessary unitsMultiply the velocity by the timeThe result should be the distance traveled in whichever length units your speed was using!