Polar to Cartesian Calculator - (r, theta) to (x, y) Solver

A polar to cartesian calculator turns a point written in polar form (r, theta) into the matching point on the rectangular plane written as (x, y), where x is the horizontal coordinate and y is the vertical coordinate. It applies x = r * cos(theta) and y = r * sin(theta) after converting the angle to radians, then surfaces x, y, the angle in degrees, and the angle in radians in a single read. Use it when a problem gives you a polar point and expects a cartesian answer, or when you need a clean (x, y) for plotting, a vector, or the rectangular form of a complex number.

• • Convert a polar point to cartesian form: A point written as (r = 10, theta = 60 degrees) becomes (x = 5, y = 8.6603) in a single step, the standard 30-60-90 right triangle in cartesian clothing.
• • Read the components of a 2D vector from a bearing: Given a length r and a heading theta, the calculator returns the horizontal and vertical components, ready to paste into a navigation, robotics, or physics calculation.
• • Convert complex numbers from polar to rectangular form: A complex number written as r * e^(i*theta) becomes the cartesian point a + bi, with a = r * cos(theta) and b = r * sin(theta).

The polar coordinate system labels every point with a distance r from the origin and an angle theta from a reference direction, while the cartesian system labels the same point with a horizontal x and a vertical y. They describe the same plane with two different sets of rulers, and the polar-to-cartesian conversion is exact.

When the input is the other way around, an (x, y) point that needs to land in (r, theta) form, Cartesian to Polar Calculator runs the inverse pair of formulas r = sqrt(x^2 + y^2) and theta = atan2(y, x) in the same style.

The calculator takes r and theta, converts theta to radians if the unit selector says degrees, and feeds the radian value into Math.cos and Math.sin to build x and y. It derives a quadrant label from the signs of x and y, and echoes the angle in both degrees and radians.

• r: Radial distance from the origin. Positive r points in the direction of theta; zero collapses it to the origin; negative r points opposite without changing theta.
• theta: Angle measured counterclockwise from the positive x-axis. The input is in degrees or radians per the unit selector; the formula always runs in radians.
• x (horizontal): Cartesian horizontal coordinate, equal to r * cos(theta_rad). Positive is right of the y-axis, negative is left.
• y (vertical): Cartesian vertical coordinate, equal to r * sin(theta_rad). Positive is above the x-axis, negative is below.

Math.cos and Math.sin in JavaScript, Python, NumPy, MATLAB, and spreadsheets expect the angle in radians, which is why the calculator applies a 180/pi conversion for degree inputs before calling cos and sin, and why the result panel always echoes theta in both units.

According to Wolfram MathWorld, the conversion from polar coordinates (r, theta) to cartesian coordinates is x = r cos(theta) and y = r sin(theta), with theta measured in radians when used inside the cosine and sine functions

According to Wikipedia (Polar coordinate system), a point given in polar form (r, theta) is converted to cartesian form (x, y) using the formulas x = r cos(theta) and y = r sin(theta), where theta is measured counterclockwise from the positive x-axis

Because x and y are just the legs of the right triangle formed by r and theta, Right Triangle Calculator is the natural place to solve for the missing leg or angle for the same input.

Four ideas cover every polar-to-cartesian conversion you will run into, from a homework problem to a robotics bearing.

The conversion is invertible: feeding (x, y) back into a cartesian-to-polar calculator gives back (r, theta) for the same point, up to the standard non-uniqueness of the polar form (any full turn added to theta is the same direction).

For a different target system such as map easting/northing, UTM, or screen pixels, Coordinates Converter is the next step in the chain once the (x, y) pair is in hand.

Five short steps take any (r, theta) input to its cartesian (x, y) answer, including the trickier cases with a negative radius or an angle in radians.

1. 1 Enter the radius r: Type the radial distance. Positive values place the point in the direction of theta; 0 collapses the point to the origin; negative r points in the opposite direction without changing theta.
2. 2 Enter the angle theta: Type the angle measured counterclockwise from the positive x-axis. Any real number is accepted; the result is taken modulo a full turn so very large angles still work.
3. 3 Pick degrees or radians: Select the unit of the theta input. The calculator converts the angle to radians before calling cos and sin, and the result is the same in both units.
4. 4 Read x and y: The top two results are the cartesian horizontal coordinate x and vertical coordinate y, ready to paste into a plot, a vector sum, or a follow-up formula.
5. 5 Check the quadrant label: The bottom result says I, II, III, IV, or the axis label (+x, -x, +y, -y), or 'Origin' when r = 0. It is the fastest way to confirm the side of the plane.

A radar reports a target at range r = 12 nautical miles and bearing theta = 215 degrees. Type r = 12, theta = 215, degrees. The x and y read -9.8296 and -6.8829 nautical miles, the horizontal and vertical offsets.

If the polar (r, theta) input is actually a complex number in polar form, Complex Number Calculator turns the same (x, y) output into a + bi with magnitude and argument.

The polar to cartesian calculator handles the four operations you would do by hand, plus the angle unit and sign cases that usually trip people up.

• • Both angle units shown in one step: The result panel echoes theta in both degrees and radians, so there is no need to remember the pi/180 or 180/pi factor mid-calculation.
• • Quadrant label built in: The calculator returns I, II, III, IV, or an axis label, so the user can confirm the side of the plane without drawing a coordinate diagram.
• • Negative radius handled correctly: A negative r applies the formula directly, placing the point in the opposite direction. This is the standard polar encoding for that case and is preserved here without rewriting the angle.
• • Origin case made explicit: When r = 0 the calculator returns x = 0, y = 0, and an 'Origin' label, instead of letting a tiny floating-point residual show up in the result.
• • Pairs with related coordinate tools: The same (x, y) feeds directly into the right triangle calculator, the complex number calculator, and the coordinates converter for a different target system.

If the next step needs the angle in gradians, DMS, or turns instead of the two units already shown, Radians to Degrees Calculator is the natural follow-up without redoing the polar-to-cartesian math.

Three things change the answer you should expect, plus two practical caveats about how the polar-to-cartesian conversion is usually defined.

• • The polar form of a non-origin point is not unique: (r, theta) and (r, theta + 360 degrees) describe the same point. The calculator reports the angle in (-180°, 180°] (or (-pi, pi] in radians).
• • The output is rounded to 4 decimal places in x and y and 6 decimal places in the radian angle, more than enough for engineering and academic work.

When the radius r is exactly zero, the angle theta is mathematically undefined and the calculator collapses the result to the origin. Treat the origin as a special case rather than zeroing out a denominator downstream.

According to Wikipedia (Trigonometric functions), the standard mathematical definition of cos(theta) and sin(theta) expects theta in radians, so any degree-based input must be multiplied by pi/180 before being passed to these functions

To check that the (x, y) output is the right length for the original r, Vector Magnitude Calculator computes sqrt(x^2 + y^2) and should return the same r the user typed in.