Expertly Reviewed by: David Chen, CFA • Updated: May 2024
Effortlessly convert between 2D coordinate systems with our polar rectangular calculator. Whether you are solving physics problems or engineering designs, this tool provides instant results and detailed mathematical steps.
Polar Rectangular Calculator
Results will appear here…
Polar Rectangular Calculator Formula
Rectangular to Polar:
$r = \sqrt{x^2 + y^2}$
$\theta = \arctan(y / x)$
Polar to Rectangular:
$x = r \cdot \cos(\theta)$
$y = r \cdot \sin(\theta)$
Formula Source: Wolfram MathWorld | MathsIsFun
Variables:
- x (Abscissa): The horizontal distance from the origin in a Cartesian plane.
- y (Ordinate): The vertical distance from the origin in a Cartesian plane.
- r (Radius): The direct distance from the origin to the point.
- θ (Theta): The angle measured counter-clockwise from the positive x-axis.
Related Calculators
What is Polar Rectangular Calculator?
A polar rectangular calculator is a specialized tool used to convert points between the Cartesian (Rectangular) coordinate system and the Polar coordinate system. In the rectangular system, a point is defined by its horizontal (x) and vertical (y) distances. In the polar system, that same point is defined by its distance from the center (r) and its angle (θ).
This conversion is vital in fields like navigation, robotics, physics, and complex number theory. Understanding how these two systems interact allows engineers to switch between the linear motion of mechanics and the rotational motion of motors and waves effortlessly.
How to Calculate Polar Rectangular Calculator (Example)
- Identify Inputs: Determine if you have (x, y) or (r, θ).
- For Rectangular to Polar: Square both x and y, sum them, and take the square root to find $r$. Use $\arctan(y/x)$ to find the angle.
- For Polar to Rectangular: Multiply $r$ by the cosine of the angle for $x$, and $r$ by the sine of the angle for $y$.
- Example: Convert Rectangular (3, 4) to Polar.
- $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.
- $\theta = \arctan(4/3) \approx 53.13^\circ$.
Frequently Asked Questions (FAQ)
Is the angle in degrees or radians? Our calculator uses Degrees by default for easier user interaction, but conversions can also be performed in Radians using $\pi$ scaling.
Can the radius (r) be negative? In standard geometry, $r$ is non-negative. However, in some advanced math, a negative $r$ signifies a point in the opposite direction of the angle.
What happens if X is zero? If $x=0$, the angle is $90^\circ$ (if $y>0$) or $270^\circ$ (if $y<0$), as the tangent is undefined at these vertical points.
Why use Polar coordinates? Polar coordinates are much more efficient for describing circles, spirals, and rotational movement compared to Rectangular coordinates.