Easily convert polar coordinates $$(r, \theta)$$ to their equivalent rectangular (Cartesian) coordinates $$(x, y)$$ using this accurate calculator. The angle $\theta$ is accepted in degrees.
Polar to Rectangular Calculator
Polar to Rectangular Calculator Formula
The conversion is based on standard trigonometric identities:
x = r * cos($\theta$)
y = r * sin($\theta$)
Variables
- Radius ($r$): The distance from the origin (pole) to the point. This value must be non-negative.
- Angle ($\theta$): The counterclockwise angle measured from the positive x-axis to the line segment connecting the point to the origin, expressed in degrees.
What is Polar to Rectangular Conversion?
In mathematics, coordinate systems are used to specify the position of a point. The two most common are the rectangular (Cartesian) system, defined by perpendicular axes $$(x, y)$$, and the polar system, defined by a distance and an angle $$(r, \theta)$$.
Converting from polar to rectangular coordinates means transforming a point’s location described by its radial distance ($r$) and its angular position ($\theta$) into the familiar horizontal ($x$) and vertical ($y$) distances. This conversion is crucial in physics, engineering, and computer graphics when moving between circular motion analysis and linear trajectory calculations.
The relationship is derived directly from the definition of sine and cosine in a right-angled triangle formed by the point, the origin, and the projection onto the x-axis. The hypotenuse is $r$, and the legs are $x$ and $y$.
How to Calculate Polar to Rectangular (Example)
- Identify the Polar Coordinates: Let’s take the polar coordinates $$(r, \theta) = (10, 60^{\circ})$$.
- Convert the Angle to Radians: The conversion formulas require radians. $Radians = 60^{\circ} \times (\pi / 180^{\circ}) \approx 1.0472$ radians.
- Calculate the X-coordinate: Use the formula $$x = r \cos(\theta)$$. $$x = 10 \times \cos(1.0472) = 10 \times 0.5 = 5$$.
- Calculate the Y-coordinate: Use the formula $$y = r \sin(\theta)$$. $$y = 10 \times \sin(1.0472) = 10 \times 0.8660 \approx 8.66$$.
- Final Result: The rectangular coordinates are $$(x, y) = (5, 8.66)$$.
Related Calculators
- Rectangular to Polar Calculator
- Vector Magnitude and Direction Calculator
- Spherical to Cartesian Converter
- Complex Number Conversion Tool
Frequently Asked Questions (FAQ)
Is the angle always measured in degrees?
While this calculator uses degrees for user input, most mathematical formulas use radians by default. The calculator automatically converts the input degrees to radians before applying the sine and cosine functions.
What is the difference between $$(r, \theta)$$ and $$(x, y)$$?
$$(r, \theta)$$ describes a point by how far it is from the center ($r$) and what angle it makes ($\theta$). $$(x, y)$$ describes a point by how far it is along the horizontal axis ($x$) and the vertical axis ($y$). They are two ways to describe the exact same location in a plane.
Can the radius ($r$) be negative?
In the basic definition of polar coordinates, $r$ represents distance and is therefore non-negative. Some advanced applications allow $r < 0$, which means moving $|r|$ units in the direction opposite to $\theta$. This calculator only accepts $r \ge 0$.
Why are these conversions important in engineering?
Conversions are vital in fields like robotics and navigation. For example, a radar system provides a target’s location in polar coordinates ($r$, $\theta$), but a machine needs to plot the target on a grid using rectangular coordinates ($x, y$) to determine pathfinding.