What Are Polar and Rectangular Coordinates?
Polar and rectangular (Cartesian) coordinates are two different ways to describe the same point in a plane. In rectangular form, you describe a point by how far it sits along the horizontal axis (x) and the vertical axis (y). In polar form, you describe the point by its distance from the origin (r) and the direction you must rotate from the positive x-axis (θ). Both descriptions represent the same geometry; they just emphasize different ideas: rectangular highlights independent horizontal/vertical components, while polar highlights distance and direction.
Understanding both systems is useful because different problems become simpler in different coordinate systems. A circle centered at the origin is complicated in rectangular form, but extremely simple in polar form: it’s just r = constant. On the other hand, adding vectors is often more direct in rectangular form because you can add x-components and y-components separately. A fast converter helps you switch representations without losing accuracy or spending time on calculator keystrokes.
How to Convert Polar to Rectangular
Polar form tells you two things: how far the point is from the origin (r), and the angle θ that points toward the location. To convert that “distance + direction” into x and y components, you use trigonometry. The cosine of the angle gives the fraction of r along the x-axis, and the sine gives the fraction along the y-axis:
x = r·cos(θ)
y = r·sin(θ)
These formulas work because you can imagine r as the hypotenuse of a right triangle formed by dropping a perpendicular to the x-axis. The adjacent side is x and the opposite side is y. Cosine relates adjacent to hypotenuse, and sine relates opposite to hypotenuse. When θ changes, the direction changes; when r changes, the point moves closer or farther away in the same direction.
How to Convert Rectangular to Polar
Converting from (x, y) back to (r, θ) means recovering distance and direction from the components. The distance from the origin is the magnitude of the vector, found with the Pythagorean theorem:
r = √(x² + y²)
The angle requires more care. A naive approach might be θ = arctan(y/x), but that fails when x is negative or when x is zero. The robust method uses the two-argument arctangent function:
θ = atan2(y, x)
atan2 uses both x and y to select the correct quadrant automatically. That matters because many different angles share the same tangent value, and a single-argument arctan cannot reliably determine whether your point is in Quadrant II, III, or IV. If you want results that match diagrams and engineering conventions, atan2 is the right tool.
Degrees vs Radians
An angle can be measured in degrees or radians. Degrees divide a circle into 360 equal parts, which is familiar and common in school geometry. Radians measure angles based on arc length, so one full rotation is 2π radians. Radians are often preferred in calculus and physics because many formulas become cleaner.
The important rule is consistency: if a problem statement or formula expects radians, entering degrees will produce incorrect results. This converter lets you choose the unit explicitly so the displayed θ matches the language of your problem. If you’re unsure, check whether the context involves π, derivatives/integrals, or angular frequency—those are strong signals that radians are intended.
What Is Angle Normalization and Why Do It?
Angles “wrap around.” A direction can be represented by infinitely many angles: 30°, 390°, and −330° all point the same way. That flexibility is mathematically correct, but it can be confusing when you want a clean, standard output. Normalization maps θ into a chosen range so results look consistent from one calculation to the next.
Two popular conventions are:
- 0 to 360° (or 0 to 2π): useful for compass-style headings and many geometry diagrams.
- −180° to 180° (or −π to π): useful when you want a signed angle that indicates rotation direction.
The key idea is that normalization changes how θ is displayed, not the underlying direction. The point remains the same.
Negative Radius in Polar Form
In many classes, r is assumed to be non-negative. In more advanced work (including complex numbers), you might see negative r values. A negative radius simply means “go in the opposite direction.” The following equivalence always holds:
(r, θ) ≡ (|r|, θ + 180°) in degrees, or (|r|, θ + π) in radians.
This converter supports negative r and will still compute the correct x and y. If you normalize θ afterward, the output will look clean and conventional.
Why Quadrants Matter
Quadrants describe where a point lies relative to the axes: Quadrant I has x > 0 and y > 0, Quadrant II has x < 0 and y > 0, Quadrant III has x < 0 and y < 0, and Quadrant IV has x > 0 and y < 0. The quadrant determines the correct angle direction.
If you only compute arctan(y/x), you can lose quadrant information because y/x can be the same for different sign combinations. atan2 fixes this by inspecting x and y separately. If you want an angle that matches the diagram on paper, always prefer atan2-style output.
Common Examples You Can Verify
If you want to sanity-check results quickly, try these:
- (x, y) = (3, 4) → r = 5, θ ≈ 53.130102°
- (x, y) = (−3, 4) → r = 5, θ ≈ 126.869898°
- (r, θ) = (5, 30°) → x ≈ 4.330127, y = 2.5
- (r, θ) = (10, 90°) → x ≈ 0, y = 10
Notice how the second rectangular example has the same r as (3, 4) but a different angle because the point is in a different quadrant.
How This Converter Helps With Complex Numbers
A complex number z = a + bi can be treated as a point (x, y) = (a, b). Its polar form is often written as r∠θ or re^{iθ}. The magnitude r is √(a² + b²), and the argument θ is atan2(b, a). Switching between these forms is common in AC circuits, control systems, signal processing, and many engineering applications because multiplication and division are often easier in polar form while addition is easier in rectangular form.
Precision, Rounding, and Real-World Measurement
This tool computes using standard floating-point math, then formats the output to the precision you select. If your inputs come from real measurements, keep in mind that physical measurement error limits how meaningful extra decimal places are. Still, adjustable precision is useful when you’re debugging formulas, matching textbook answers, or preparing values for a spreadsheet or report.
What If Your Result Looks “Wrong”?
Most “wrong result” moments come from one of three issues:
- Unit mismatch: using degrees when the formula expects radians (or the other way around).
- Normalization expectations: your textbook may prefer −180° to 180° while your output is 0 to 360°.
- Sign/quadrant confusion: forgetting that x and y signs affect θ, especially in Quadrants II and III.
Use the normalization selector to match the style you want, and rely on atan2-based angles for correct quadrant selection.
FAQ
Polar ↔ Rectangular Converter – Frequently Asked Questions
Answers about formulas, units, normalization, quadrants, and common mistakes when converting coordinates.
Rectangular (Cartesian) form describes a point as (x, y). Polar form describes the same point as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis.
Use x = r·cos(θ) and y = r·sin(θ). Make sure θ is in the correct unit (degrees or radians) for your calculation.
Use r = √(x² + y²) and θ = atan2(y, x). atan2 returns the correct quadrant for θ, which is important when x or y is negative.
Degrees are common in geometry and many everyday problems. Radians are common in calculus, physics, engineering, and complex-number work. Choose the unit that matches your formula or textbook.
Angles can be represented in multiple equivalent ways. For example, 30°, 390°, and −330° point in the same direction. Angle normalization (like 0–360° or −180–180°) changes the “presentation,” not the direction.
Normalization maps θ into a standard range, such as 0–360° (or 0–2π) or −180–180° (or −π–π). This makes results easier to compare and reduces confusion in repeated calculations.
A negative radius can be converted by flipping the direction: (r, θ) is equivalent to (|r|, θ + 180°) in degrees (or θ + π in radians). This tool supports negative r and will still compute the correct x and y.
atan2(y, x) computes the angle using both x and y so it can determine the correct quadrant automatically. Using atan(y/x) alone can give incorrect angles when x is negative or when x = 0.
Yes. A complex number a + bi matches rectangular form (x, y). Its polar form is r∠θ where r is the magnitude and θ is the argument. The same conversion formulas apply.
No. Calculations run in your browser. Nothing is saved to a server or database.