Goal
Convert polar (r, degrees) to rectangular (x, y) and back.
Polar Coordinate Converter
Convert polar (r, degrees) to rectangular (x, y) and back.
Key facts
- What it does
- Convert polar (r, degrees) to rectangular (x, y) and back.
- Formula
x = r cos(theta); y = r sin(theta); r = sqrt(x^2+y^2); theta = atan2(y,x).- You enter
- Direction · Radius r (or x) · Angle in degrees (or y)
- Worked example
- x = 3, y = 4.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Direction · Radius r (or x) · Angle in degrees (or y)
Method
x = r cos(theta); y = r sin(theta); r = sqrt(x^2+y^2); theta = atan2(y,x).
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Polar Coordinate Converter
Convert polar (r, degrees) to rectangular (x, y) and back.
Open the Polar Coordinate Converter pageMore math tools
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Result
Enter your values above and choose Calculate to see the result here.
Calculation map
Follow the path from input to answer
01
Inputs (3)
- Direction Ready
- Radius r (or x) Ready
- Angle in degrees (or y) Ready
02
Formula
x = r cos(theta); y = r sin(theta); r = sqrt(x^2+y^2); theta = atan2(y,x).
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
Recent runs
Your recent runs stay in this browser session only.
Formula, assumptions, and example
Formula: x = r cos(theta); y = r sin(theta); r = sqrt(x^2+y^2); theta = atan2(y,x).
Polar input uses degrees with the positive x-axis at zero. Reverse conversion uses atan2 so quadrants resolve, normalized to 0-360 degrees.
- Angles are degrees, not radians.
- Reverse angle is normalized to 0-360 degrees; origin gives 0.
Worked example: x = 3, y = 4.
Displayed input contract
- Direction · 2 choices
- Radius r (or x) · minimum -1000000000000 · maximum 1000000000000
- Angle in degrees (or y) · minimum -1000000000000 · maximum 1000000000000
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Polar Coordinate Converter for a real question
Convert polar (r, degrees) to rectangular (x, y) and back. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes polar rectangular, r theta convert, atan2 degrees. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Direction · Radius r (or x) · Angle in degrees (or y). Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. Angles are degrees, not radians.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Polar Coordinate Converter
- Enter Direction.
- Enter Radius r (or x) — As r for polar input; as x for rectangular input.
- Enter Angle in degrees (or y) — As degrees for polar input; as y for rectangular input.
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
x = r cos(theta); y = r sin(theta); r = sqrt(x^2+y^2); theta = atan2(y,x).
Polar input uses degrees with the positive x-axis at zero. Reverse conversion uses atan2 so quadrants resolve, normalized to 0-360 degrees.
Worked example
x = 3, y = 4.
Assumptions and limits
- Angles are degrees, not radians.
- Reverse angle is normalized to 0-360 degrees; origin gives 0.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
A point in a plane can be described by its horizontal and vertical coordinates or by a distance from the origin and a direction. This calculator converts between those rectangular and polar descriptions. In polar-to-rectangular mode, enter a radius r and an angle in degrees to obtain x and y. In rectangular-to-polar mode, enter x and y to obtain the nonnegative radius and an angle normalized to the interval from 0 through 360 degrees. The formulas are x = r cos(theta), y = r sin(theta), r = sqrt(x squared + y squared), and theta = atan2(y, x). The page accepts negative radii in the forward direction because a directed polar segment can point opposite the stated angle. It does not silently convert degrees to radians in the input contract, and it does not claim that one point has only one possible polar representation. This guide explains the two modes, quadrant behavior, negative radius, origin handling, validation, rounding, reversibility, and the model limits of a point converter. The output is transparent coordinate arithmetic, not a graph, route, measurement, or engineering recommendation.
Two ways to name one point
Rectangular coordinates write a point as (x, y). The x-value measures signed horizontal displacement from the origin, and the y-value measures signed vertical displacement. Polar coordinates write a point as (r, theta), where r is a directed distance from the origin and theta is measured from the positive x-axis. Both systems describe location in the same plane, but they organize the information differently. Rectangular form is often convenient for algebraic equations, while polar form can make distance and direction easier to express.
The two descriptions are not two different physical points when they convert correctly. They are coordinate representations tied to the same chosen axes and origin. A conversion result should therefore be checked by converting it back or by plotting both descriptions on the same coordinate system. The calculator does not draw that plot, so the meaning of positive direction, axis orientation, and units comes from the problem in which the result is used.
- Rectangular form is (x, y).
- Polar form is (r, theta).
- Both forms describe location relative to the same origin and axes.
- A conversion changes notation, not the underlying point.
Choose the direction before entering values
The Direction selector has two choices: Polar to rectangular and Rectangular to polar. In the first choice, the first numeric field is interpreted as radius r and the second as an angle in degrees. In the second choice, the first numeric field is interpreted as x and the second as y. The labels include both meanings so that switching the selector does not require guessing which field changed role.
The same pair of numeric values can produce different results after switching modes because the meanings change. For example, first = 5 and second = 53.13 describes a radius and angle in forward mode, but x = 5 and y = 53.13 in reverse mode. This is expected. Select the mode first, read the field hints, and then enter values with the selected interpretation.
- Polar-to-rectangular interprets first as r and second as degrees.
- Rectangular-to-polar interprets first as x and second as y.
- The selector is part of the calculation contract.
- Switching modes changes field meaning, not just the label.
Angles are entered in degrees
The forward mode expects the angle field in degrees. The handler converts that value to radians internally by multiplying by pi and dividing by 180 before calling the JavaScript sine and cosine functions. You should therefore enter 90 for a right-angle direction, not pi divided by 2, unless you have first converted the angle yourself. Entering a radian value while thinking it is degrees creates a valid but different direction.
The forward angle is not normalized before evaluation. An angle of 450 degrees is accepted and points in the same direction as 90 degrees because sine and cosine are periodic. An angle of -30 degrees is also accepted and points below the positive x-axis. Preserving the entered angle in this direction can be useful when a problem records a rotation history, even though many angles represent the same final ray.
- Enter degrees, not radians, in the angle field.
- 90 degrees points along the positive y-axis.
- Negative and larger-than-360-degree angles are allowed within bounds.
- Forward mode preserves the direction mathematically without normalizing the input.
Polar to rectangular formulas
To convert (r, theta) to (x, y), resolve the directed radius into horizontal and vertical components. The horizontal component is x = r cos(theta), and the vertical component is y = r sin(theta). The trigonometric functions use the internally converted radian angle, while the user-facing field remains in degrees. These equations come from the right-triangle relationships between the radius and the two rectangular components, extended with signs to cover all quadrants.
The radius is multiplied into both components. A radius of 5 at 0 degrees gives x = 5 and y = 0. At 90 degrees it gives x = 0 and y = 5, up to floating-point rounding. At 180 degrees it gives x = -5 and y = 0. These axis cases are good mental checks because one component should vanish and the other should carry the radius with the expected sign.
- x = r cos(theta).
- y = r sin(theta).
- The theta used by JavaScript trigonometry is converted from degrees.
- Axis angles provide quick sign and zero checks.
The default example
The default forward inputs are r = 5 and theta = 53.13 degrees. The page calculates x = 5 cos(53.13 degrees) and y = 5 sin(53.13 degrees). Since 53.13 degrees is a rounded angle whose ideal three-four-five triangle angle is about 53.1301 degrees, the displayed components are close to x = 3 and y = 4. The catalog example describes the result as x = 3, y = 4 because the values are intended as a readable rounded demonstration.
The conversion is not using a special three-four-five shortcut. It evaluates the trigonometric expressions for the entered decimal angle, then formats the numeric outputs to six decimal places. A user who enters a more precise angle may see slightly different final digits. That difference is not a contradiction; it reflects the precision of the angle supplied. For a strict exact triangle, enter a known exact axis angle or carry the angle precision required by the problem.
- Default r is 5 and default angle is 53.13 degrees.
- The result is approximately x = 3 and y = 4.
- The handler evaluates sine and cosine rather than recognizing a triangle pattern.
- The displayed result is rounded for presentation.
Quadrants control component signs
With a positive radius, angles from 0 to 90 degrees place a point in the first quadrant, where x and y are both nonnegative. Angles from 90 to 180 degrees generally place the point in the second quadrant, where x is negative and y is positive. From 180 to 270 degrees both components are negative, and from 270 to 360 degrees x is positive while y is negative. The calculator follows the signs returned by cosine and sine rather than applying a quadrant label itself.
At an axis, one component is zero. The exact mathematical value may be represented by a tiny floating-point residue, which the result formatter rounds for display. If a result near an axis shows a very small value, compare its magnitude with the displayed precision before treating it as meaningful displacement. For a negative radius, the component signs are additionally reversed relative to the same positive-radius angle, which is why radius sign and quadrant should be considered together.
- Quadrant I has positive x and positive y for positive r.
- Quadrant II has negative x and positive y.
- Quadrant III has negative x and negative y.
- Quadrant IV has positive x and negative y.
Negative radius is a directed segment
The forward radius field may be negative because the handler permits values from -1e12 through 1e12. A negative r does not mean that distance itself is negative in ordinary geometric language. It means the directed segment travels in the direction opposite the stated angle. Algebraically, (-r, theta) describes the same point as (r, theta + 180 degrees), when both representations are interpreted in the same coordinate system.
For example, r = -2 and theta = 30 degrees gives x = -2 cos(30 degrees) and y = -2 sin(30 degrees), placing the point opposite the 30-degree ray. This is a valid polar representation, not an input error. The reverse conversion does not preserve a negative radius: it computes the nonnegative distance from x and y and reports an equivalent normalized direction.
- Negative r reverses the direction of the ray.
- (-r, theta) is equivalent to (r, theta + 180 degrees).
- Negative radius is accepted only within the numeric input bounds.
- Reverse mode returns a nonnegative radius.
Rectangular to polar radius
In reverse mode, the radius is calculated as r = sqrt(x squared + y squared). The implementation uses the hypotenuse operation so that the two components are combined as a Euclidean distance. This radius is never negative. It measures how far the point lies from the origin, regardless of its quadrant or the signs of x and y.
For x = 3 and y = 4, the result is r = sqrt(9 + 16) = 5. For x = -3 and y = -4, the same radius is returned because distance does not depend on direction. A very large coordinate pair can still produce a finite radius within the handler's result guard, but the number's practical units and scale must come from the surrounding problem. The page does not convert meters, pixels, or other units.
- r = sqrt(x squared + y squared).
- The reverse radius is always zero or positive.
- Opposite points have the same distance from the origin.
- No physical or display-unit conversion is performed.
Why atan2 is used for the angle
A one-argument inverse tangent can lose quadrant information because a ratio y divided by x is the same for points on opposite rays. The handler uses atan2(y, x), which considers both signed components and returns the direction of the vector. This distinguishes, for example, (1, 1) from (-1, -1), even though both ratios y/x equal 1.
The raw atan2 angle is converted from radians to degrees. If it is negative, 360 degrees is added once so the returned angle lies from 0 up to but not including 360 for non-origin points. This convention makes a point below the x-axis readable as a positive clockwise-equivalent position in the selected range. It is a representation choice; negative angles and angles above 360 can describe the same direction.
- atan2 uses both x and y signs.
- It resolves the correct quadrant.
- The result is converted to degrees.
- Negative reverse angles are shifted into the 0-360 convention.
Angle normalization in reverse mode
For a point in the fourth quadrant, such as x = 1 and y = -1, atan2 returns a negative angle near -45 degrees. The calculator adds 360 and reports 315 degrees. The normalized value points in exactly the same direction but fits the stated output convention. For a point in the second quadrant, the raw angle is already positive, so no addition is needed. The normalization occurs after direction has been resolved, not before.
The range convention is useful for comparing outputs because equivalent negative directions are placed in one common cycle. It does not remove the fact that polar coordinates are nonunique. A point can also be written with an angle plus or minus any full turn, and a negative radius can shift the angle by 180 degrees. The reported pair is one canonical choice for this page, not the only valid pair in mathematics.
- Fourth-quadrant directions become angles between 270 and 360 degrees.
- No angle below zero is returned in reverse mode.
- Adding 360 preserves the ray direction.
- Canonical output does not imply unique polar representation.
The origin is a special case
When x = 0 and y = 0, the radius is zero. Direction at the origin is not geometrically unique because every angle starts from the same point and a zero-length segment never leaves it. The handler chooses angle 0 degrees for this case so the output is deterministic and easy to report. This is a convention rather than a discovery of a preferred direction.
The forward direction also accepts r = 0. Both x and y then calculate to zero for every finite angle. If a worksheet asks for the angle of the origin, explain that the angle is undefined or arbitrary mathematically, while noting that this calculator displays the selected convention only when converting the rectangular origin to polar form. Do not infer a physical heading from a zero vector.
- The rectangular origin converts to radius 0.
- The handler reports origin angle 0 degrees by convention.
- A zero vector has no unique direction.
- Forward r = 0 produces x = 0 and y = 0.
Validation and numeric boundaries
The direction selector must be one of the two known option values. Both numeric fields must be actual finite JavaScript numbers within the inclusive range from -1e12 to 1e12. The handler rejects strings, missing values, NaN, positive or negative infinity, and numbers beyond the limit. These checks remain in the pure engine so direct callers receive the same behavior as the browser form.
The bounds are computational guardrails, not a claim that a trillion-unit radius or angle is automatically useful. A huge angle can be converted to a large radian value and still produce finite sine and cosine values, but the last digits may be sensitive to floating-point argument reduction. A large coordinate can be mathematically valid while being inappropriate for a measured map or instrument. Check the domain and units separately.
- Mode must be a recognized conversion option.
- Both numeric inputs are finite and bounded.
- The page rejects coercion-prone values such as numeric strings.
- Accepted range is not a domain-specific measurement guarantee.
Precision and displayed results
The handler returns numeric x, y, radius, and angle values with six-place display precision. The result object retains numbers so the shared renderer can format them consistently. Rounding is a presentation step. It can make an axis residue appear as 0 or make a near-three-four-five conversion read as 3 and 4, but it does not change the underlying floating-point value used by a caller.
Trigonometric calculations are subject to ordinary floating-point limits. Decimal degree inputs are approximations, and common angles may not be represented exactly after conversion to radians. If a result is near a threshold, a boundary, or a geometric intersection, carry the unrounded values in a separate calculation and define the tolerance. Do not treat six displayed decimal places as six guaranteed significant digits of the original measurement.
- Numeric results are displayed to six decimal places.
- Rounding can hide tiny axis residues.
- Input angle precision limits coordinate precision.
- Use an explicit tolerance for boundary decisions.
Check a conversion in both directions
A practical verification is to convert a result back. Start with r and theta, calculate x and y, then use those components in reverse mode. The radius should return close to the magnitude of the original r, but if the original radius was negative, the reverse radius will be its absolute magnitude and the angle will be shifted to an equivalent direction. Likewise, a forward angle may return a normalized angle after the round trip.
For a positive-radius point, compare x and y with the original direction using a tolerance appropriate to the displayed precision. For the default example, the reverse result from approximately (3, 4) is a radius near 5 and an angle near 53.13 degrees. Small differences are expected because the catalog example uses a rounded decimal angle. A round trip is a consistency check, not proof that the inputs came from a correct real-world coordinate system.
- Forward then reverse conversion should preserve the point within tolerance.
- Negative r returns as a nonnegative radius with an equivalent angle.
- Reverse angles may be normalized from a negative representation.
- Round trips test arithmetic consistency, not source-data accuracy.
Units and practical coordinate use
The coordinate formulas preserve the length unit attached to r, x, and y. If r is in meters, x and y are in meters. If x and y are pixels, the returned radius is in pixels. The angle remains an angle and is reported in degrees in this page's contract. No scale factor, map projection, compass convention, or latitude-longitude interpretation is added automatically.
For a drawing, a robot sketch, a vector exercise, or a local map, define where the origin lies and which way the positive axes point. A compass bearing often uses a different zero direction and clockwise-positive convention than a mathematical angle from the positive x-axis. Convert that convention before using this page. Do not paste geographic latitude and longitude into the fields and assume the result is a geodesic position; geographic coordinates require a reference surface and a different model.
- Output length units match the entered length units.
- The mathematical angle starts at the positive x-axis.
- Compass bearings and geographic coordinates need separate conventions.
- The page performs no map scale or projection conversion.
What this converter does not decide
The calculator converts one coordinate pair at a time. It does not plot a curve, transform an equation, calculate a path length, solve an intersection, or determine whether a point lies inside a region. It also does not infer a coordinate reference system, sensor calibration, map projection, image origin, or physical axis orientation. Those details may change how a mathematically correct pair should be used.
It does not turn a polar pair into a unique physical direction when r is zero, and it does not determine which equivalent polar representation a textbook or application prefers. It does not attach uncertainty intervals or decide how many decimal places a survey, drawing, or measurement requires. Use the page for the stated trigonometric conversion, then add the coordinate-system rules and accuracy evidence required by the larger task.
- No graph or equation transformation is performed.
- No geographic projection or compass convention is inferred.
- No uncertainty or calibration model is included.
- One valid representation is not the only mathematically valid representation.
A repeatable reporting workflow
Write the point, origin, axis orientation, length unit, and angle convention before opening the calculator. Select the direction that matches the values you have. In forward mode, verify that the radius and degree angle are paired. In reverse mode, verify that the first and second values are x and y rather than a radius and angle. Check signs with a rough quadrant sketch before pressing Calculate.
After calculation, record the mode, raw inputs, formula, displayed outputs, and rounding rule. If the result will be reused, retain more precision than the visual card shows and apply a stated tolerance. For an important workflow, run the reverse conversion and compare the point. If a result is used in a physical or geographic system, document the external coordinate convention and calibration instead of presenting the coordinate pair as self-explanatory.
- Document origin, axes, units, and angle convention.
- Confirm field roles after selecting the mode.
- Check quadrant signs before interpreting the output.
- Record raw inputs, displayed precision, and any round-trip check.
The useful boundary of the model
This calculator is a compact way to connect right-triangle trigonometry with coordinate geometry. It is useful for checking component resolution, converting a vector between common forms, preparing a point for a graphing exercise, and explaining why atan2 is safer than a simple inverse tangent for quadrant-aware work. The model is intentionally small enough that each result can be audited from four elementary formulas.
Its reliability comes from keeping the contract visible: degrees in the input angle, signed components, directed forward radius, nonnegative reverse radius, normalized reverse angle, explicit origin convention, bounded finite inputs, and six-place display formatting. Use those statements when teaching or documenting the result. A clear limitation is part of a correct coordinate conversion, not an apology for the arithmetic.
- Use it for transparent point and vector coordinate conversion.
- Keep angle units and axis conventions visible.
- Use a round trip or independent formula as a check when needed.
- Do not extend a point converter into an unprovided physical model.
Frequently asked questions
What is the Polar Coordinate Converter?
Convert polar (r, degrees) to rectangular (x, y) and back.
What is the formula for the Polar Coordinate Converter?
x = r cos(theta); y = r sin(theta); r = sqrt(x^2+y^2); theta = atan2(y,x). Polar input uses degrees with the positive x-axis at zero. Reverse conversion uses atan2 so quadrants resolve, normalized to 0-360 degrees.
What do I need to use this calculator?
Enter Direction, Radius r (or x), Angle in degrees (or y), then choose Calculate.
What are the limits of this calculator?
Angles are degrees, not radians. Reverse angle is normalized to 0-360 degrees; origin gives 0.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.