polar to rectangular calculator

If you have a vector, point, or complex number written in polar form and need standard Cartesian coordinates, this calculator converts it instantly. Enter the magnitude r, the angle θ, choose degrees or radians, and get x and y values right away.

How polar to rectangular conversion works

Polar form describes a point by distance from the origin and direction. Rectangular form describes the same point by horizontal and vertical coordinates. The conversion formulas are:

• x = r cos(θ)
• y = r sin(θ)

Where r is the radius (magnitude) and θ is the angle measured from the positive x-axis.

Quick example

Example in degrees

Suppose you have r = 10 and θ = 30°. Then:

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

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

Example in radians

If r = 4 and θ = 1.2 radians, the same formulas apply directly without unit conversion.

Degrees vs radians

Many calculation errors come from angle-unit mismatch. If your angle is in degrees, the calculator converts it internally to radians before applying trigonometric functions. If your angle is already in radians, no conversion is needed.

• Use degrees for geometry and basic coordinate problems.
• Use radians for calculus, physics, and most programming formulas.

Where this conversion is used

• Complex numbers: Convert r(cosθ + i sinθ) into a + bi.
• Engineering vectors: Resolve force and velocity into components.
• Signal processing: Move between phasor and Cartesian representation.
• Computer graphics: Position points using angle and radius, then render with x-y coordinates.

Common mistakes to avoid

• Entering degrees while radians are selected (or vice versa).
• Rounding too early during intermediate steps.
• Forgetting that negative radius values reflect direction.
• Expecting exact integers when trig values are irrational.

Tips for accurate results

Use enough decimal places

For engineering or physics work, keep at least 4 to 6 decimal places until your final answer.

Check with known angles

For sanity checks, try familiar inputs like 45°, 60°, 90°, or π/2.

Interpret signs carefully

Signs of x and y indicate the quadrant, which helps verify whether the output matches your angle.