Polar to Cartesian Coordinates
Enter a polar coordinate (r, θ) and choose whether θ is in degrees or radians. The calculator applies x = r·cos(θ) and y = r·sin(θ) to get the Cartesian coordinates, with the degree-to-radian conversion shown when needed.
The direction with no ambiguity
Of the two conversions between polar and rectangular coordinates, this is the easy one. Going from a distance and a direction to a pair of coordinates is two multiplications, with no decisions to make along the way — the answer is determined, and there is never a second candidate to choose between.
This calculator applies those two multiplications, accepting the angle in either degrees or radians and showing the conversion when one is needed.
The reverse conversion has to decide which quadrant a point lies in, and it produces an angle that is only one of infinitely many describing the same direction. Nothing like that arises here: every polar pair names exactly one point, and cosine and sine deliver it without a case analysis.
That makes this the safer direction to compute in when a choice exists. Where a calculation can be arranged to convert this way rather than the other, no convention has to be agreed on and no quadrant can be got wrong.
How to use this calculator
- Enter the radius The distance from the origin along the given direction. It may be negative, which is a legitimate convention rather than an error.
- Enter the angle Measured anticlockwise from the positive horizontal axis. Any value is accepted, including negatives and angles beyond a full turn.
- Choose the unit Degrees or radians. Getting this wrong is the one way to produce a badly wrong answer here, since both units accept the same numbers.
- Check the conversion line when working in degrees The radian equivalent is shown before the trigonometry is applied, which confirms the unit selector matches what you intended.
The formula, and where it comes from
x = r·cos θ y = r·sin θ θ_rad = θ_deg × π/180
The two formulas are the definitions of cosine and sine applied to a right triangle whose hypotenuse is the radius. The horizontal leg is the radius times the cosine and the vertical leg the radius times the sine, which is all the conversion consists of.
Because both coordinates come from the same radius and the same angle, they are consistent by construction — the resulting point is always exactly the given distance from the origin. That is a property the reverse conversion has to work to preserve and this one gets for free.
The degree conversion is applied first when it is needed, and it is reported as its own step. The trigonometric functions work in radians internally, so the multiplication by π over 180 is not a display convenience but a necessary part of the calculation.
A negative radius is handled by the arithmetic rather than by a special case. Multiplying by a negative flips the signs of both coordinates, which places the point exactly opposite the direction the angle names — the standard reading of a negative radius.
What each input means
- r Radius — form field “Radius r”
- Distance from the origin along the direction given. A negative value reflects the point through the origin rather than being invalid.
- θ Angle — form field “Angle θ”
- Measured anticlockwise from the positive horizontal axis. Adding a full turn leaves the point unchanged.
- unit Angle unit — form field “Angle unit”
- Degrees or radians. It changes what the angle number means, and nothing else about the calculation.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A 3-4-5 triangle in reverse
A radius of 5 at an angle whose tangent is four thirds. The result should be the point three across and four up.
Inputs Radius r = 5, Angle θ = 53.130102, Angle unit = deg
- Polar coordinates (r, θ) = (5, 53.1301°)
- Convert to radians 53.1301° × π/180 = 0.927295 rad
- x x = r·cos(θ) = 5 × 0.6 = 3
- y y = r·sin(θ) = 5 × 0.8 = 4
Result (x, y) = (3, 4)
The angle is entered as a decimal approximation rather than an exact value, so the coordinates come back as 3 and 4 only to the precision of that decimal. This is a case where the input, not the method, limits the accuracy.
Squaring and adding the two coordinates returns 25, the radius squared. That check works on every conversion of this kind and confirms the pair is consistent with the distance asked for.
An angle in the second quadrant
A radius of 2 at 135 degrees. Past a right angle, so the horizontal coordinate turns negative while the vertical stays positive.
Inputs Radius r = 2, Angle θ = 135, Angle unit = deg
- Polar coordinates (r, θ) = (2, 135°)
- Convert to radians 135° × π/180 = 2.35619 rad
- x x = r·cos(θ) = 2 × -0.707107 = -1.41421
- y y = r·sin(θ) = 2 × 0.707107 = 1.41421
Result (x, y) = (-1.41421, 1.41421)
The cosine is negative here and the sine positive, which places the point up and to the left. The signs come out of the trigonometric functions themselves — no quadrant rule is applied on top.
The two coordinates are equal in magnitude, since the angle is exactly midway between the vertical and horizontal directions. Angles that are multiples of 45 degrees always produce that symmetry.
The same calculation in radians
A radius of 2 at a third of π, entered as a decimal with the unit set to radians.
Inputs Radius r = 2, Angle θ = 1.0471975512, Angle unit = rad
- Polar coordinates (r, θ) = (2, 1.0472 rad)
- x x = r·cos(θ) = 2 × 0.5 = 1
- y y = r·sin(θ) = 2 × 0.866025 = 1.73205
Result (x, y) = (1, 1.73205)
No conversion line appears, because none is needed — the value is already in the unit the trigonometric functions use. That absence is the confirmation the selector was set as intended.
Entering the same number with the unit left on degrees would give a completely different point, roughly one degree from the horizontal axis. The two units accept identical input, so nothing but the selector prevents the mistake.
Reading the result
Every polar pair names one point
The conversion is a genuine function: one radius and one angle give one result, always. The reverse is not, which is why converting back may return a different-looking pair that describes the same location.
A negative radius reflects through the origin
It does not make the point invalid or the distance meaningless. The pair with a negative radius describes the same location as one with the positive radius and the angle turned by half a turn.
Angles repeat every full turn
Adding or subtracting a complete revolution leaves the point exactly where it was, so an angle of 380 degrees and one of 20 give identical coordinates. There is no need to reduce an angle before entering it.
When you would use this
Plotting a polar equation by hand
Evaluating a polar function at several angles gives a list of pairs, and converting each produces the points to plot on ordinary axes — exactly what a polar grapher does at every sample.
Resolving a magnitude and bearing into components
A force or velocity given as a size and direction becomes a pair of components through this conversion, which is the form needed before several can be added together.
Assumptions and limitations
What this calculator assumes
- The angle is measured anticlockwise from the positive horizontal axis.
- Degrees are converted to radians before the trigonometric functions are applied.
- A negative radius is interpreted as a reflection through the origin rather than rejected.
- Both coordinates derive from the same radius and angle, so the result is always at the stated distance.
Where it stops being the right tool
- Two dimensions only: cylindrical and spherical coordinates are not covered.
- One point at a time, so a list of polar pairs must be converted individually.
- The angle is taken from the positive horizontal axis anticlockwise; a compass bearing convention needs converting first.
- Results are decimal, so a coordinate that should be an exact surd is shown rounded.
Common mistakes
Leaving the unit selector on the wrong setting
Why it happens. Both units accept the same numbers without complaint. A value of 1.05 is a plausible radian angle and an equally plausible degree one, and the two land in completely different places.
How to avoid it. Look for the conversion line. It appears only in degree mode, so its presence or absence tells you which unit was used before the answer is read.
Swapping the cosine and the sine
Why it happens. Both appear in symmetric-looking formulas, and neither position is obviously the right one when recalled from memory rather than derived.
How to avoid it. Anchor on a right angle. At 90 degrees the point is straight up, so the vertical coordinate must carry the sine, which is 1 there while the cosine is 0.
Rejecting a negative radius
Why it happens. A distance cannot be negative in ordinary usage, so a negative value looks like an input error to be corrected.
How to avoid it. Enter it as given. The convention places the point on the opposite ray, and several standard polar curves rely on it.
Frequently asked questions
Can the radius be negative?
Yes. A negative radius reflects the point through the origin, so it describes the same location as the positive radius with the angle increased by half a turn. The arithmetic handles it without a special case.
What units should I use?
Either — the selector chooses between degrees and radians, and the value is converted internally before cosine and sine are applied. In degree mode the radian equivalent is shown as its own step.
Is this the inverse of the Cartesian-to-polar tool?
Yes, and this is the well-behaved direction of the pair. Every polar input names exactly one point, whereas the reverse has to choose a quadrant and returns one of infinitely many equivalent angles.
Does an angle beyond a full turn need reducing first?
No. Cosine and sine repeat every full revolution, so 380 degrees and 20 degrees give identical coordinates. Negative angles are equally acceptable and simply measure clockwise.