# Polar to Rectangular Calculator

Convert between polar (r, theta) and rectangular (x, y) coordinates. Supports degrees and radians. Uses x = r cos(theta), y = r sin(theta).

## What this calculates

Convert between polar and rectangular coordinate systems in either direction. Enter polar coordinates (r, theta) to get (x, y), or enter rectangular coordinates (x, y) to get (r, theta). For example, the polar point (5, 53.13 degrees) converts to approximately (3, 4) in rectangular form.

## Inputs

- **Conversion Direction** — options: Polar to Rectangular, Rectangular to Polar — Choose which direction to convert.
- **Angle Unit** — options: Degrees, Radians — The unit for the angle.
- **First Value** — For polar to rectangular: enter r. For rectangular to polar: enter x.
- **Second Value** — For polar to rectangular: enter theta. For rectangular to polar: enter y.

## Outputs

- **x (Rectangular)** — The x-coordinate.
- **y (Rectangular)** — The y-coordinate.
- **r (Polar)** — The distance from the origin.
- **Theta (degrees)** — The angle in degrees.
- **Theta (radians)** — The angle in radians.

## Details

**Polar to Rectangular Conversion**

Given polar coordinates (r, theta):
- **x = r x cos(theta)**
- **y = r x sin(theta)**

Example: Convert (10, 60 degrees) to rectangular.
- x = 10 x cos(60) = 10 x 0.5 = 5
- y = 10 x sin(60) = 10 x 0.866 = 8.66
- Result: (5, 8.66)

**Rectangular to Polar Conversion**

Given rectangular coordinates (x, y):
- **r = sqrt(x^2 + y^2)**
- **theta = atan2(y, x)**

Example: Convert (3, 4) to polar.
- r = sqrt(9 + 16) = sqrt(25) = 5
- theta = atan2(4, 3) = 53.13 degrees
- Result: (5, 53.13 degrees)

**Degrees vs. Radians**

This calculator supports both angle units. To convert between them:
- Degrees to radians: multiply by pi/180
- Radians to degrees: multiply by 180/pi

Common angles: 90 degrees = pi/2 radians, 180 degrees = pi radians, 360 degrees = 2pi radians.

**When to Use Each System**

- **Rectangular (x, y):** Best for straight lines, linear algebra, and Cartesian problems
- **Polar (r, theta):** Best for circles, spirals, rotations, and problems with radial symmetry

## Frequently Asked Questions

**Q: What are polar coordinates?**

A: Polar coordinates describe a point using its distance from the origin (r) and the angle from the positive x-axis (theta). They are an alternative to the rectangular (Cartesian) system of (x, y). Polar coordinates are natural for describing circles, spirals, and rotational phenomena.

**Q: How do you convert polar to rectangular?**

A: Multiply the radius by cosine of the angle to get x, and by sine of the angle to get y. The formulas are x = r cos(theta) and y = r sin(theta). Make sure your angle is in the correct unit (degrees or radians) before computing.

**Q: Why does the calculator use atan2 instead of atan?**

A: The atan function only returns angles in the range -90 to 90 degrees, so it cannot distinguish between all four quadrants. The atan2(y, x) function considers the signs of both x and y to return the correct angle in the full range of -180 to 180 degrees.

**Q: Can r be negative in polar coordinates?**

A: In some conventions, yes. A negative r means you go in the opposite direction from the angle. For instance, (-5, 0 degrees) is the same point as (5, 180 degrees). This calculator expects non-negative r values, which is the most common convention.

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Source: https://vastcalc.com/calculators/math/polar-rectangular
Category: Math
Last updated: 2026-04-08