Rectangular & Polar Coordinates Converter

In mathematics and engineering, coordinate systems are essential tools for representing points in space. Two of the most commonly used coordinate systems are the rectangular (Cartesian) coordinate system and the polar coordinate system. Each has its own advantages and applications, and being able to convert between them is crucial in many fields such as physics, engineering, computer graphics, and navigation.



Rectangular and Polar Coordinates Converter is an online tool that enables you to seamlessly switch between and convert coordinates in these two systems.

What Is the Rectangular (Cartesian) Coordinate System?

The rectangular coordinate system, also known as the Cartesian coordinate system, was introduced by French mathematician René Descartes. It uses two perpendicular axes—usually labeled x and y—to locate a point on a plane. A point is represented as an ordered pair (x, y), where:

• x is the horizontal distance from the origin.

• y is the vertical distance from the origin.

This system is particularly useful for problems involving straight lines, rectangles, and grid-based structures. It forms the basis of analytic geometry and is widely used in algebra, calculus, and engineering design.

What Is the Polar Coordinate System?

The polar coordinate system represents a point in a plane using a distance and an angle relative to a fixed reference point (the origin) and a fixed reference direction (usually the positive x-axis). A point is expressed as (r, θ), where:

• r is the radial distance from the origin to the point.

• θ (theta) is the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point.

Polar coordinates are especially useful in scenarios involving circular or rotational symmetry, such as in physics (e.g., describing orbits), signal processing, and complex number theory.

Converting Between Rectangular and Polar Coordinates

From Rectangular to Polar:

To convert a point (x, y) from rectangular to polar coordinates:

• $r=\sqrt{x^2+y^2}$

• $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Note: The value of θ must be adjusted based on the quadrant in which the point lies to ensure the correct angle.

From Polar to Rectangular:

To convert a point (r, θ) from polar to rectangular coordinates:

• $x=r\cdot \cos (\theta )$

• $y=r\cdot \sin (\theta )$

These conversions are fundamental in many mathematical and technical applications, including vector analysis, control systems, and even GPS technology.

Rectangular and Polar Coordinates Converter is a tool designed to automate the conversion process between these two coordinate systems. It allow you to input values in one coordinate system and instantly receive the corresponding values in the other.