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The Distance Formula

Updated on October 10, 2015

Mathematics (an exact science)

Mathematics | Source

What is Mathematics?

Mathematics is the science that treats the study of quantity, shape, space and change. There is a wide scope of disciplines into which mathematics fits in the everyday world. Mathematics makes use of reason and logic to arrive to conclusions, therefore, this discipline was and is one of the most exact branches of human knowledge. Without mathematics, the world would not have advanced into modernity.

The science of numbers has been practiced; as far back as human records exist. Humanity has utilized mathematics to keep a record of their businesses and nowadays, math is used in disciplines, including astronomy, engineering, architecture, finance and many more. It is one of the most difficult academic subjects to dominate; nevertheless, it is one of the most exact.

Scientists make use of mathematical models to relate to and understand real world phenomena. Many disciplines rely on the discoveries made by mathematicians to develop new disciplines in the areas of technology, science and engineering.

Measuring Distances in the Universe

The Universe
The Universe | Source

Distances on Earth and the Cosmos

In the everyday world is very useful to know the quantities, either to keep personal records or to know the distances from one place to another. Scientists have been able to calculate with an almost exact approximation the number of stars and galaxies in the universe; moreover, with the use of numbers, they have calculated the number of cells in the human body and the distances in the cosmos.

To calculate distances on earth, we make use of the metric system, so to measure the distance from one pace to another, we use the kilometer or mile in the US. To measure distances in the cosmos, scientists make use of more sophisticated methods, including the parallax, Cepheid’s variables, redshift calculation and a super novae explosion.

Many times, it may go unnoticed, but if we knew the distance from the top of a building to the bottom of the floor, we would be able to know the time it would take an object to fall from its top to the bottom.

Throughout this hub, I´m going to show you how to calculate the distance from a point to another using the distance formula.

The Distance Formula

Distance Formula
Distance Formula | Source

The Cartesian Coordinate System

To locate two points in the Cartesian coordinate system, we assign two coordinates to a point. One coordinate goes from the origin to either the left or right horizontal direction or the other coordinate goes from the origin in a top or down direction.

The X-axis represents the horizontal coordinates and the Y-axis represents the vertical coordinates. The X-axis measures the distance left or right from the origin and the Y-axis measure the distance up or down from the origin.

Distance Between Two Points

To locate the distance in the cartesion coordinate system, we specify two points of coordinates. the lines runing horizontally are called the X axis and the ones running vertically are called the Y axis. The point where the Y and X intersects it´s the origin.

The numbers in the horizontal and vertical axis indicate the coordinates in the carftesian plane. The origin has a value of 0 on both the X axis and the Y axis. The points extending to the right from the origin are positive and so are the points extnding up from the origin. The points extending to the left of the oringin are negative and so are the one extending to the bottom from the origin.

For any given point, we assign its corresponding numbers; for example, let´s assign numbers to two points in the coordinate system. First we assign the numbers 3 and 4 to a point and then assign -3 and 2 to another point.


Coordinates in the Cartesian Plane

To locate these two points, we plot them on the Cartesian plane by placing a dot at its location. Now, we connect these two points with a straight line. Finally, we introduce these coordinates into the distance formula.

And the result is the square root of 40 which is equal to the real number 6.3245.

Two Points in the Cartesian Plane


The Pythagorean Theorem

By the Pythagorean Theorem, the distance between two points is about the same. The theorem states that the square of the hypotenuse in a straight triangle is equal to the sum of the squares of the other two sides.

So, if we draw a horizontal line from point (3.2) to point (-3, 2) and then a vertical line from point (3, 2) to point (3.4) we have the figure of a straight triangle. Now, to find the square of the hypotenuse, which is the longest side on the triangle, we plot the shorter sides in the formula.


Practical Uses of the Distance Formula

As its name implies, the distance formula can be used to measure distances. The distance formula can be utilized in Google maps to know the distances from a region to another; for example, if we wanted to know the distance from an area in Russia to another in Mexico, we just draw a triangle on the map and use the formulas presented here to calculate the distances.

Measuring the Distance in a Map of the World Using the Distance Formula



The distance formula is more effective when measuring distances on flat surfaces. To get an exact measure of the distance from one city to another city in the world, you will have to consider that the earth is round and use another type of trigonometric calculation.


Submit a Comment

  • unvrso profile image

    Jose Juan Gutierrez 23 months ago from Mexico City

    That is true! To get an exact measure of the distance from Russia to Mexico or to another city in the world, you will have to consider that the earth is round and use another type of trigonometric calculation.

    For the purpose of the topic, It´s important to say that this measure is just an approximation.

    Thanks for reading and commenting!

  • BlossomSB profile image

    Bronwen Scott-Branagan 23 months ago from Victoria, Australia

    I'm blinded by science, or rather, by mathematics. Sounds interesting, but beyond me. I don't understand how you can measure distances on a flat map when the earth is round.