Effortlessly convert Cartesian coordinates (x, y) to Polar coordinates (radius r, angle θ) with our high-precision rectangular to polar calculator.
Rectangular to Polar Calculator
Rectangular to Polar Calculator Formula
To convert from Rectangular $(x, y)$ to Polar $(r, \theta)$, we use the following mathematical transformations:
θ = atan2(y, x)
Formula Source: Wolfram MathWorld – Polar Coordinates | Khan Academy Reference
Variables:
- x (Horizontal Axis): The directed distance from the y-axis to the point.
- y (Vertical Axis): The directed distance from the x-axis to the point.
- r (Radius/Magnitude): The distance from the origin $(0,0)$ to the point.
- θ (Theta/Angle): The angle measured counter-clockwise from the positive x-axis.
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What is a Rectangular to Polar Calculator?
A Rectangular to Polar Calculator is a specialized tool used to transform coordinates from the Cartesian system (based on a grid of horizontal and vertical lines) into the Polar system (based on distance from a center and angle from a reference direction). This is a fundamental operation in trigonometry, calculus, and engineering.
While Cartesian coordinates define a point by “how far right and how far up,” Polar coordinates define it by “how far away and at what angle.” This is particularly useful in physics for describing rotational motion, sound waves, and electromagnetic fields.
How to Calculate Rectangular to Polar (Example)
Suppose you have a point with coordinates x = 3 and y = 4.
- Calculate Radius (r): Square both numbers ($3^2=9$, $4^2=16$), add them ($9+16=25$), and take the square root ($\sqrt{25} = 5$).
- Calculate Angle (θ): Use the inverse tangent function ($\arctan(y/x)$). For (3, 4), $\arctan(4/3) \approx 53.13^\circ$.
- Final Result: The polar form is $(5, 53.13^\circ)$.
Frequently Asked Questions (FAQ)
If $x=0$, the simple $\arctan(y/x)$ formula fails due to division by zero. However, our calculator uses the `atan2` function, which correctly handles these cases (e.g., if $x=0, y=5$, the angle is $90^\circ$).
Standard calculators provide both. Our tool displays the angle in degrees for easy visualization and in radians for mathematical calculations.
In standard geometric conversions, the radius $r$ is always non-negative ($r \ge 0$), representing a physical distance from the origin.
The calculator automatically identifies the quadrant (I, II, III, or IV) based on the signs of $x$ and $y$ to provide the correct angle between $0^\circ$ and $360^\circ$.