polar to rectangular form calculator

How this polar to rectangular form calculator works

Complex numbers can be written in two common formats: polar form and rectangular form. Polar form emphasizes a number’s distance from the origin (magnitude) and direction (angle), while rectangular form emphasizes horizontal and vertical components on the complex plane.

This calculator helps you move from polar notation r∠θ to rectangular notation a + bi in one step. It is useful in algebra, trigonometry, electrical engineering, signals, controls, and physics.

Core conversion formula

If a complex number is given as z = r∠θ, then:

Where:

• r is the magnitude (also called modulus).
• θ is the angle (argument), in degrees or radians.
• a is the real part.
• b is the imaginary coefficient.

Example conversion

Convert 10∠45° to rectangular form

• Real part: a = 10 cos(45°) ≈ 7.0711
• Imaginary part: b = 10 sin(45°) ≈ 7.0711
• Rectangular result: z ≈ 7.0711 + 7.0711i

That is exactly the same operation the calculator performs behind the scenes.

Why both forms matter

Each form has strengths:

• Polar form is great for multiplication, division, roots, and powers.
• Rectangular form is convenient for addition, subtraction, and plotting x/y components.

In practice, people often switch between forms repeatedly depending on the operation being performed.

Degrees vs radians (important)

One of the most common mistakes in complex-number calculations is mixing angle units. This calculator includes a unit selector so you can choose the correct interpretation.

• Use degrees if your angle is like 30°, 90°, 225°, etc.
• Use radians if your angle is like π/6, 1.2, 3.14, etc.

If you enter a degree value while radians are selected, your answer will be incorrect.

Practical applications

1) AC circuit analysis

Voltages and currents are often represented as phasors in polar form. To add or subtract them, you commonly convert to rectangular first.

2) Control systems and signals

Frequency-response values are frequently in magnitude-angle format. Converting to real and imaginary components enables deeper analysis and plotting.

3) Mathematics education

Students use this conversion to understand how trigonometry connects to complex numbers and coordinate geometry.

Common errors to avoid

• Using degrees when the input angle is in radians (or vice versa).
• Forgetting the sign of the imaginary part when the sine value is negative.
• Rounding too early in multi-step problems.
• Assuming the rectangular form must always have both nonzero parts.

Quick interpretation tips

• If θ = 0°, the number lies on the positive real axis.
• If θ = 90°, the number is purely imaginary and positive.
• If θ = 180°, the number is real and negative.
• If θ = 270°, the number is purely imaginary and negative.

FAQ

Can r be negative?

In standard polar form for complex numbers, magnitude is usually nonnegative. Mathematically, a negative r still produces a valid result through the same formulas, and this calculator supports it.

Does this calculator normalize the angle?

No normalization is required to compute rectangular coordinates. Since sine and cosine are periodic, any equivalent angle gives the same rectangular point.

How many decimals should I use?

For homework, 3–4 decimals are often enough. For engineering use, choose a precision level that matches your measurement or simulation requirements.