How do you convert polar to rectangular coordinates?

Use x = r·cos(θ) and y = r·sin(θ). The angle θ determines direction from positive x-axis, and r is the distance from origin.

How do you convert rectangular to polar coordinates?

Use r = √(x² + y²) for distance and θ = arctan(y/x) for angle, adjusting θ based on the quadrant (signs of x and y).

Why are polar coordinates useful?

They simplify problems with circular symmetry. Circles become r = constant, spirals have simple equations, and angular motion is naturally described.

How do quadrants affect polar angle calculation?

arctan gives angles only in Q1 and Q4. For Q2 and Q3, add π (180°). Check signs: Q1(+,+), Q2(-,+), Q3(-,-), Q4(+,-).

Polar to Rectangular Calculator

Convert between polar coordinates (r, θ) and rectangular coordinates (x, y) with step-by-step solutions.

Radius (r)

Angle (θ)

Angle Unit

Conversion Formulas:

Polar to Rectangular:

x = r × cos(θ)

y = r × sin(θ)

Rectangular to Polar:

r = √(x² + y²)

θ = atan2(y, x)

Common Angle Reference

Degrees	Radians	cos(θ)	sin(θ)
0°	0	1	0
30°	π/6	√3/2	1/2
45°	π/4	√2/2	√2/2
60°	π/3	1/2	√3/2
90°	π/2	0	1
180°	π	-1	0
270°	3π/2	0	-1
360°	2π	1	0

Understanding Coordinate Systems

Rectangular (Cartesian)

•Uses x (horizontal) and y (vertical) distances
•Best for linear motion and grids
•Origin at intersection of axes

Polar

•Uses distance (r) and angle (θ) from origin
•Best for circular/rotational motion
•θ measured counterclockwise from x-axis

Related Math Tools

About Polar to Rectangular Calculator - Coordinate Converter

Convert between polar (r, θ) and rectangular (x, y) coordinates. Supports degrees and radians with step-by-step conversion formulas.

Our **Polar to Rectangular Calculator** converts between coordinate systems instantly. Enter polar coordinates (r, θ) or rectangular coordinates (x, y) and get the equivalent representation with step-by-step formulas. For complex number conversions, see our Imaginary Number Calculator.

Polar coordinates use distance from origin (r) and angle from positive x-axis (θ), while rectangular coordinates use horizontal (x) and vertical (y) distances. Converting between them is essential for calculus, physics, and engineering applications.

Our calculator supports both degrees and radians, shows all conversion steps, and handles all quadrants correctly. Understand the geometric relationship between the two systems with our visual explanations.

Key Features

Polar to rectangular conversion

Rectangular to polar conversion

Degree and radian modes

All quadrants supported

Step-by-step formulas

Exact values when possible

Why Use This Tool?

Simplify calculus problems

Physics applications

Navigation calculations

Engineering analysis

Graphing polar equations

Common Use Cases

Calculus: Evaluate polar integrals and derivatives.

Physics: Motion in circular paths.

Navigation: Radar and direction finding.

Complex Numbers: Rectangular to polar form.

Related Tools

Imaginary Number Calculator

Complex number operations.

Line Integral Calculator

Path integrals in any coordinates.

How to Use

Select conversion direction

Choose degrees or radians

Enter your coordinates

Click Convert

View results and formulas