polar rectangular calculator

What this polar rectangular calculator does

This calculator converts a point between the two most common 2D coordinate systems: rectangular coordinates (x, y) and polar coordinates (r, θ). If you are working in trigonometry, calculus, vectors, signal processing, or physics, you will often switch between these forms.

Use polar to rectangular when you know distance and direction from the origin. Use rectangular to polar when you know horizontal and vertical components and need magnitude and angle.

How to use the calculator

• Select the conversion direction from the dropdown.
• Enter your known values in the input boxes.
• Choose degrees or radians when entering a polar angle.
• Click Calculate to view converted values and formulas used.
• Click Clear to reset all fields quickly.

Core formulas

Polar to rectangular

These equations project the radius onto the horizontal and vertical axes. Make sure θ is in radians when evaluating sine and cosine in most programming contexts.

Rectangular to polar

The atan2 function is essential because it determines the correct quadrant. A plain arctangent of y/x can produce incorrect angles whenever x is negative or zero.

Worked examples

Example 1: Convert (r=10, θ=30°) to rectangular

Convert angle to radians only when needed by your tool: 30° = π/6. Then apply formulas:

• x = 10 cos(30°) ≈ 8.6603
• y = 10 sin(30°) = 5

So the rectangular form is approximately (8.6603, 5).

Example 2: Convert (x=-3, y=4) to polar

• r = √((-3)² + 4²) = √25 = 5
• θ = atan2(4, -3) ≈ 126.87° (Quadrant II)

So the polar form is (5, 126.87°) or (5, 2.2143 rad).

Common mistakes to avoid

• Mixing degrees and radians: always verify unit settings before calculating.
• Using arctan(y/x) instead of atan2(y, x): this causes wrong quadrant angles.
• Rounding too early: keep extra precision in intermediate steps.
• Ignoring equivalent angles: 30°, 390°, and -330° describe the same direction.

Where these conversions are used

Polar-rectangular conversion appears in many technical fields:

• Complex numbers: convert between a + bi and re^iθ.
• Physics: represent forces, velocity vectors, and wave motion.
• Engineering: AC circuit phasors and control systems.
• Computer graphics and robotics: angle-based movement and targeting.

FAQ

Can radius be negative in polar form?

Yes. A negative radius points in the opposite direction of the given angle. The calculator handles this mathematically and still returns valid rectangular coordinates.

Why does the same point have multiple polar angles?

Angles are periodic. Adding or subtracting 360° (or 2π radians) points in the same direction, so infinitely many angle values can represent one location.

Should I choose principal or normalized angle output?

Use principal angle when you want the default atan2 output (negative angles allowed). Use normalized output when your class or project requires angles from 0° to 360°.

Bottom line

A fast polar rectangular calculator saves time and reduces errors, especially when solving repeated coordinate conversion problems. Enter your values, verify angle units, and use the results directly in your homework, code, or engineering workflow.