Our website is an independent, advertising-supported platform. We provide our content free of charge to our readers, and to keep it that way, we rely on revenue generated through advertisements and affiliate partnerships. This means that when you click on certain links on our site and make a purchase, we may earn a commission. Learn more.
How We Make Money
We sustain our operations through affiliate commissions and advertising. If you click on an affiliate link and make a purchase, we may receive a commission from the merchant at no additional cost to you. We also display advertisements on our website, which help generate revenue to support our work and keep our content free for readers. Our editorial team operates independently of our advertising and affiliate partnerships to ensure that our content remains unbiased and focused on providing you with the best information and recommendations based on thorough research and honest evaluations. To remain transparent, we’ve provided a list of our current affiliate partners here.
Science

## Our Promise to you

Founded in 2002, our company has been a trusted resource for readers seeking informative and engaging content. Our dedication to quality remains unwavering—and will never change. We follow a strict editorial policy, ensuring that our content is authored by highly qualified professionals and edited by subject matter experts. This guarantees that everything we publish is objective, accurate, and trustworthy.

Over the years, we've refined our approach to cover a wide range of topics, providing readers with reliable and practical advice to enhance their knowledge and skills. That's why millions of readers turn to us each year. Join us in celebrating the joy of learning, guided by standards you can trust.

# What are Polar Coordinates?

By Jeffrey L. Callicott
Updated: May 21, 2024
Views: 14,148

Polar coordinates are a form of expressing position on a two-dimensional plane. Cartesian coordinates, also called rectangular coordinates, utilize a distance in each of two dimensions to locate a point, but polar coordinates make use of an angle and a distance. The distance is sometimes referred to as the radius.

Rectangular coordinates typically are denoted (x,y), where x and y are distances along those respective axes. In a similar manner, polar coordinates are expressed as (r,θ). The letter r is the distance from the origin at the angle represented by the Greek letter theta, θ, where r can be a positive or negative number. If a negative distance is used, the magnitude of the distance does not change, but the direction is taken opposite the angle θ on the other side of the origin. A point in a polar coordinate system can be referred to as representing a vector, with a magnitude of r, a direction of θ and a sense of direction, which is the sign of r.

Translation between rectangular and polar coordinates can be accomplished through the use of trigonometric formulas. For conversion from rectangular to polar, the following formulas can be applied: θ = tan-1(y/x) and r = √(x2 + y2). For changes from polar to rectangular, these equations can be employed: x = rcosθ and y = rsinθ.

Polar coordinates tend to be used for any situation in which rectangular coordinates would prove difficult or awkward to utilize, and vice versa. Any application involving circular geometry or radial movement is ideally suited to polar coordinates, because these geometries can be described with relatively simple equations in a polar coordinate system; their graphs are more curvilinear or circular in appearance compared to those on rectangular coordinate systems. As a result, polar coordinates have use representing models of real-world phenomena that have similarly rounded shapes.

The applications of polar coordinates are quite varied. Polar coordinate graphs have been used to model the sound fields produced by varying speaker locations or the areas where different types of microphones can best pick up sound. Polar coordinates are of great importance modeling orbital motions in astronomy and space travel. They are also the graphical basis for the famous Euler Formula, which is regularly applied in mathematics for representation and manipulation of complex numbers.

Like their rectangular counterparts, polar coordinates need not be limited to only two dimensions. To express values in three dimensions, a second angle represented by the Greek letter phi, φ, can be added to the coordinate system. Any point can be thus located from the origin by a fixed distance and two angles, and it can be assigned the coordinates (r,θ,φ). When this type of nomenclature is used for tracking and locating points in three dimensional space, the coordinate system is designated as a spherical coordinate system. This type of geometry is sometimes referred to as using polar spherical coordinates.

Spherical coordinates actually have a well-known application — they are used in mapping the Earth. The angle θ is typically the latitude and is limited to between minus-90 degrees and 90 degrees, whereas the angle φ is longitude and is held to between minus-180 and 180 degrees. In this application, r can sometimes be ignored, but it is more often employed for the expression of elevation above mean sea level.

All The Science is dedicated to providing accurate and trustworthy information. We carefully select reputable sources and employ a rigorous fact-checking process to maintain the highest standards. To learn more about our commitment to accuracy, read our editorial process.