Free Polar to Rectangular Coordinates Calculator

r (radius)

θ (angle)

x = r · cos(θ) y = r · sin(θ)

Enter polar coordinates r and θ to convert to rectangular coordinates.

Understanding Polar and Rectangular Coordinates

In mathematics, specifying a point on a two‑dimensional plane can be done either with polar coordinates or rectangular (Cartesian) coordinates. The polar system represents a point by its distance $r$ from the origin and the angle $\phi$ that the radius vector makes with the positive horizontal axis. On the other hand, the rectangular system uses the horizontal coordinate $x$ and vertical coordinate $y$ to denote a point’s location. Both systems are equally valid, and converting between them is a common requirement in geometry, physics, and engineering.

The Conversion Formula

To convert polar coordinates $(r, \phi)$ to rectangular coordinates $(x, y)$ , apply these simple trigonometric relations:

x = r \cos \phi \qquad y = r \sin \phi

The formulas are derived from the right triangle formed by dropping a perpendicular from the point to the $x$ -axis.
Manual calculation involves evaluating the sine and cosine of $\phi$ , which can be error‑prone—especially when the angle is not one of the standard values.

Using This Polar to Rectangular Coordinates Calculator

This free online coordinate conversion calculator takes care of all the trigonometry for you. To obtain rectangular coordinates from polar:

Enter the radial coordinate $r$ – the distance from the origin. It must be zero or positive.
Enter the angular coordinate $\phi$ – the angle from the positive x‑axis. You can input it in radians or degrees, depending on the calculator’s mode selection.
Read the results – the calculator instantly displays the corresponding $x$ and $y$ values.

Using the tool eliminates manual calculation and reduces the chance of mistakes, making it an efficient way to convert polar to Cartesian coordinates.

Example

Suppose you want to find the rectangular coordinates for the point with polar coordinates $(7, 60^\circ)$ . First, note that $60^\circ = \pi/3$ radians. Applying the conversion formulas:

$x = 7 \cos 60^\circ = 7 \times 0.5 = 3.5$
$y = 7 \sin 60^\circ = 7 \times \frac{\sqrt{3}}{2} \approx 6.062$

Entering $r = 7$ and $\phi = 60$ (in degree mode) into this polar to rectangular coordinates calculator will return the same pair $(3.5, 6.062)$ in seconds.

Whether you’re studying coordinate geometry or solving practical problems, the tool provides quick, reliable results for converting polar coordinates to rectangular.

FAQ

1. What is the formula for converting polar to rectangular coordinates?

The conversion uses x = r cos φ and y = r sin φ, where r is the radial distance and φ is the angle from the positive x‑axis.

2. Can I enter the angle in degrees instead of radians?

Yes, the calculator allows you to choose the angle unit. Make sure you select degrees or radians appropriately before entering the angle.

3. What happens if I enter a negative value for r?

The radial coordinate r must be non‑negative in the polar system. If you enter a negative value, the calculator may not produce a valid conversion because the definition of polar coordinates assumes r ≥ 0.

4. Is there any difference between polar coordinates and Cartesian coordinates?

Yes, they are two ways to describe a point on a plane. Polar coordinates use distance and angle (r, φ), while rectangular coordinates use horizontal and vertical offsets (x, y). Both systems can represent the same point, but the conversion formulas are applied accordingly.

5. How does the tool handle the trigonometry for me?

The calculator automatically evaluates the sine and cosine of the given angle, then multiplies them by r to produce x and y. You do not need to compute these trigonometric functions manually.

How to Use

Enter the radius r and the angle θ of the point in polar coordinates.
Select the angle unit - degrees, radians, gradians, turns, or π radians.
The calculator instantly displays the rectangular coordinates (x, y) using the formulas x = r cos θ and y = r sin θ.