Polar and Rectangular Coordinates Converter (r and θ, Quadrant Steps and a Graph)

Choose the direction of the conversion and enter the numbers. For "rectangular → polar", enter the x- and y-coordinates. For "polar → rectangular", enter the radius r and the angle θ (and choose its unit).

Conversion

x-coordinate

y-coordinate

Radius r

Angle θ

Unit of the angle

Enter numbers only. Decimals, negative numbers and fractions such as 3/4 are OK. A blank x or y counts as 0.

Result and figure

Enter the numbers in the fields on the left and press "Calculate". The result and a figure of the coordinate plane will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Convert rectangular coordinates \(P(x,\ y)\) to polar coordinates \(P(r,\ \theta)\) on the spot. The radius is given as an exact value with a simplified root, such as \(\sqrt{8} = 2\sqrt{2}\), and for special angles the angle is given exactly as a fraction of π
The angle \(\theta\) is given mainly in the range \(0 \le \theta < 2\pi\), with the value for \(-\pi < \theta \le \pi\) shown too, so you can check your answer whichever range your textbook uses
You can also convert the other way (polar → rectangular). Enter the radius \(r\) and the angle \(\theta\) (in degrees, as a fraction of π, or in radians as a decimal), and you get \(x = r\cos\theta\) and \(y = r\sin\theta\) as exact values such as \(\left(-2,\ 2\sqrt{3}\right)\)
The steps show, one at a time, how the angle is decided from \(\tan\theta = \dfrac{y}{x}\) and the quadrant the point is in (the quadrant adjustment)
You can check the result on a figure of the coordinate plane. Light concentric circles (a scale for the radius) and rays (a scale for the angle) show at a glance where the radius \(r\) and the angle \(\theta\) are

This page covers polar coordinates in the plane, as taught in Precalculus. For rectangular \(x\) and \(y\), you can enter fractions and decimals (rational numbers), but not irrational numbers such as √3. With such inputs, the angle comes out as an exact fraction of π only when the point is on an axis or at a 45°-family angle. (For 30° and 60° angles, tanθ is irrational, like 1/√3, so they cannot come from rational \(x\) and \(y\).) In the other direction, polar → rectangular, every special angle that is a multiple of 30° or 45° is calculated exactly.

What is this calculation used for?

Air traffic control and ship radar (location by distance and bearing)

Airport radar and ship radar measure "the distance from here" by the time a radio pulse takes to bounce back, and "the bearing" by the direction the antenna is pointing. So what they measure is polar coordinates \(\left(r,\ \theta\right)\) from the start.
But the controller's screen and nautical charts are built on horizontal and vertical coordinates, so inside the equipment the measured distance and bearing are converted to rectangular coordinates with \(x = r\cos\theta\) and \(y = r\sin\theta\). This page's conversion is what connects a device that measures by distance and direction with a screen that draws across and up.

Distance sensors in self-driving cars and robot vacuums

The laser distance sensors (LiDAR) on self-driving cars and robot vacuums spin around and take a huge number of measurements of the form "something reflected the beam this far away in this direction". Each measurement is a pair of direction (angle) and distance, which is polar coordinates.
To build a map from the collected points, every point is converted to rectangular coordinates and then combined. Mapping the surroundings and then planning a route is the basic flow of self-driving, and this coordinate conversion is where it starts.

Weather radar and hurricane tracking maps

Weather radar also spins its antenna while sending out radio waves, and receives the reflections from rain and snow as "bearing and distance". The rain images in TV weather forecasts are this polar data converted to the horizontal and vertical coordinates of a map.
Giving the position of a hurricane as "150 miles south-southeast of Miami" is also the polar way of thinking, and the conversion to rectangular coordinates happens when the point is plotted on a map.

Designing machines and rides that go around in circles

For things that turn around a center, such as the cars on a Ferris wheel, the teeth of a gear or the blades of a fan, design gets simpler if you think "the distance from the center is fixed and only the direction changes". In polar coordinates you just fix the distance \(r\) and change the angle \(\theta\), which makes placing the parts very easy.
But drawings and CNC machine tools take horizontal and vertical coordinates, so at the end every position is converted to rectangular coordinates before it is handed over.

Circular layouts and motion in computer graphics and games

Placing enemies evenly around the hero, firing bullets in all directions, orbiting a camera around a target: these are most natural to think of in terms of "distance from the center" and "angle", and splitting the angle evenly decides the layout.
The screen is drawn in horizontal and vertical coordinates, so every frame the positions are converted with \(x = r\cos\theta\) and \(y = r\sin\theta\). This conversion is why circular motion is easy to program.

Formulas and figures

From polar to rectangular (two ways to give the location of a point)

Figure

Standard notation (the usual math form)

\(x\) \(=\) \(r\) \(\cos\theta\)

\(y\) \(=\) \(r\) \(\sin\theta\)

In words (symbols replaced with words)

③ \(x\)-coordinate \(=\) ① \(r\): radius ② cosine of the angle \(\theta\)

④ \(y\)-coordinate \(=\) \(r\): radius sine of the angle \(\theta\)

The formula in words

① Take the distance from the pole (the origin), the radius \(r\)

② multiply it by the cosine of the angle \(\theta\), \(\cos\theta\)

③ and you get the \(x\)-coordinate

④ Multiply the same radius by the sine \(\sin\theta\), and you get the \(y\)-coordinate

Quick example

Converting the polar coordinates \(P\left(2,\ \dfrac{\pi}{3}\right)\) (distance 2 from the pole, direction 60° from the polar axis) to rectangular coordinates gives

\(x\)-coordinate \(=\) radius \(2\) \(\cos 60^{\circ} = \dfrac{1}{2}\)

\(y\)-coordinate \(=\) radius \(2\) \(\sin 60^{\circ} = \dfrac{\sqrt{3}}{2}\)

\(x = 2\cos\dfrac{\pi}{3} = 2 \times \dfrac{1}{2} = 1\)

\(y = 2\sin\dfrac{\pi}{3} = 2 \times \dfrac{\sqrt{3}}{2} = \sqrt{3}\)

\(P\left(2,\ \dfrac{\pi}{3}\right) \rightarrow P\left(1,\ \sqrt{3}\right)\)

Key idea

These are two ways to describe the location of the same point. "Go \(x\) to the right and \(y\) up" is rectangular coordinates \((x,\ y)\). "Go the distance \(r\) from a reference point, in the direction turned \(\theta\) from a reference direction" is polar coordinates \((r,\ \theta)\). When meeting a friend, it is the difference between "300 feet east and 400 feet north of the station" and "500 feet from the station, roughly to the northeast". The reference point of polar coordinates is called the pole, and the reference direction (the ray from the pole to the right) is called the polar axis. When you lay polar coordinates over the coordinate plane, the pole goes on the origin and the polar axis on the positive \(x\)-axis. To see why \(x = r\cos\theta\) and \(y = r\sin\theta\), think of the circle of radius 1 (the unit circle). On the unit circle, the point in the direction of the angle \(\theta\) has horizontal position \(\cos\theta\) and vertical position \(\sin\theta\). If the radius is \(r\), the whole picture is scaled up \(r\) times, so both positions are multiplied by \(r\).

Radius \(r\) (distance from the pole)

Standard notation (the usual math form)

In words (symbols replaced with words)

\(r\) \(=\) \(\sqrt{x^{2} + y^{2}}\)

② \(r\): radius \(=\) ① square root of the sum of the squares of \(x\) and \(y\)

The formula in words

① The square root of \(x\) squared plus \(y\) squared, \(\sqrt{x^2 + y^2}\),

② is the radius \(r\)

Quick example

The radius of the point \(P\left(3,\ 4\right)\) is

radius \(r\) \(=\) \(\sqrt{3^2 + 4^2}\)

\(r = \sqrt{3^{2} + 4^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Key idea

This is exactly the Pythagorean theorem from Grade 8. Join the origin, the point \((x,\ 0)\) and \(P\), and you get a right triangle with legs \(x\) and \(y\). The distance from the origin to \(P\) (the hypotenuse) is \(\sqrt{x^2 + y^2}\). The root in the answer is usually written in simplified form, with square factors taken out, as in \(\sqrt{8} = 2\sqrt{2}\). This calculator also shows the simplified form. In polar coordinates, the radius \(r\) is taken to be 0 or greater. A distance is never negative, so the \(r\) you get from the conversion is never negative.

Angle \(\theta\) (from tanθ and the quadrant)

Standard notation (the usual math form)

In words (symbols replaced with words)

\(\tan\theta\) \(=\) \(\dfrac{y}{x}\)

② tangent of the angle \(\theta\) \(=\) ① \(y\) divided by \(x\)

The formula in words

① The \(y\)-coordinate divided by the \(x\)-coordinate

② is the tangent of the angle \(\theta\), \(\tan\theta\) (this alone does not fix \(\theta\), so you narrow it down by the quadrant that \(P\) is in)

Quick example

The angle of the point \(P\left(-1,\ 1\right)\) (\(x\) is negative and \(y\) is positive, so the point is in Quadrant II) is

tangent \(\tan\theta\) \(=\) \(\dfrac{1}{-1} = -1\)

\(\tan\theta = \dfrac{1}{-1} = -1\)

\(\theta = \dfrac{3}{4}\pi \quad \left(x < 0,\ y > 0\right)\)

Key idea

In the range \(0 \le \theta < 2\pi\), there are two angles with \(\tan\theta = -1\): \(\dfrac{3}{4}\pi\) (135°) and \(\dfrac{7}{4}\pi\) (315°). Which one is right depends on the quadrant of the point \(P\). \(P(-1,\ 1)\) has negative \(x\) and positive \(y\), so it is in Quadrant II (upper left), and we choose \(\theta = \dfrac{3}{4}\pi\). This is the quadrant adjustment. The \(\arctan\) (inverse tangent) on a calculator or in a spreadsheet only returns angles between \(-\dfrac{\pi}{2}\) and \(\dfrac{\pi}{2}\), so for points in Quadrants II and III it does not give the angle directly. You need to adjust the angle from \(\arctan\), for example by adding \(\pi\) depending on the quadrant. (This calculator shows this adjustment in the steps.) For a point with \(x = 0\) (on the \(y\)-axis), \(\dfrac{y}{x}\) cannot be calculated. In that case, decide from the figure that \(\theta = \dfrac{\pi}{2}\) or \(\dfrac{3}{2}\pi\). If you want a method that works for every point, choose the \(\theta\) that satisfies both \(\cos\theta = \dfrac{x}{r}\) and \(\sin\theta = \dfrac{y}{r}\). Then you do not need to split into quadrant cases.

The same point has more than one set of polar coordinates

Standard notation (the usual math form)

\(P\left(r,\ \theta\right)\) \(=\) \(P\left(r,\ \theta + 2n\pi\right)\)

In words (symbols replaced with words)

① point \(P\) at polar coordinates \(\left(r,\ \theta\right)\) \(=\) ② point with \(n\) full turns \(2\pi\) added to the angle

The formula in words

① The point \(P\) at polar coordinates \(\left(r,\ \theta\right)\)

② and the point with \(n\) full turns \(2\pi\) added to the angle are the same point (\(n\) is an integer)

Quick example

Other polar coordinates for the same point as \(P\left(2,\ \dfrac{\pi}{3}\right)\) are

\(P\left(2,\ \dfrac{\pi}{3}\right)\) \(=\) \(P\left(2,\ \dfrac{7}{3}\pi\right)\) \(=\) \(P\left(2,\ -\dfrac{5}{3}\pi\right)\)

\(\dfrac{\pi}{3} + 2\pi = \dfrac{\pi}{3} + \dfrac{6}{3}\pi = \dfrac{7}{3}\pi\)

\(\dfrac{\pi}{3} - 2\pi = \dfrac{\pi}{3} - \dfrac{6}{3}\pi = -\dfrac{5}{3}\pi\)

Key idea

In rectangular coordinates, each point has exactly one pair \((x,\ y)\), but polar coordinates are different. After one full turn (\(2\pi\) = 360°) you face the same direction again, so adding or subtracting \(2\pi\) to the angle any number of times gives the same point. Since the answer is not fixed as it is, you agree on a range when you give an answer. The two common ones are \(0 \le \theta < 2\pi\) (0° or more and less than 360°) and \(-\pi < \theta \le \pi\) (more than −180° and up to 180°), and textbooks differ on which they use. This calculator shows both answers. The pole (the origin) is special. Its distance is 0, so \(r = 0\), but it has no direction, so the angle is not fixed. The polar coordinates of the pole are written \(\left(0,\ \theta\right)\) (\(\theta\) is any value), usually \(\left(0,\ 0\right)\).

A point in the plane can be written in rectangular coordinates \(\left(x,\ y\right)\) (horizontal and vertical position) or in polar coordinates \(\left(r,\ \theta\right)\) (distance from the pole and direction from the polar axis). From polar to rectangular, use \(x = r\cos\theta\) and \(y = r\sin\theta\). The other way, use \(r = \sqrt{x^2 + y^2}\) and \(\tan\theta = \dfrac{y}{x}\), plus the quadrant adjustment that picks the angle by the quadrant of the point. An angle shifted by full turns gives the same point, so answers are given in a fixed range such as \(0 \le \theta < 2\pi\).

Symbols and terms

Symbols

\(P\)	pee	A letter often used for a point in the plane, from the first letter of "point". The coordinates of the point are written in the parentheses after it, as in \(P\left(1,\ 2\right)\).
\(O\)	oh	The letter for the origin (the pole), from the first letter of "origin". In polar coordinates, distances are measured from this point.
\(x,\ y\)	x, y	The horizontal (\(x\)-axis direction) and vertical (\(y\)-axis direction) positions in rectangular coordinates. Both can be negative.
\(r\)	are	The letter for the radius (the distance from the pole to the point), from the first letter of "radius". In polar coordinates it is a real number 0 or greater.
\(\theta\)	theta	The Greek letter used for the angle. In math it is the usual letter for an angle.
\(\left(r,\ \theta\right)\)	r, theta	How polar coordinates are written: first the distance (radius), then the direction (angle). It looks just like rectangular \(\left(x,\ y\right)\), so always say which kind of coordinates you are using.
\(\cos\theta,\ \sin\theta\)	cosine theta, sine theta	Trigonometric ratios (trigonometric functions). On the circle of radius 1 (the unit circle), the point in the direction of the angle \(\theta\) has horizontal position \(\cos\theta\) and vertical position \(\sin\theta\). They are the key to converting polar to rectangular.
\(\tan\theta\)	tangent theta	One of the trigonometric ratios, defined as \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\). On the coordinate plane, \(\tan\theta = \dfrac{y}{x}\) (rise over run, the slope of the line through the origin and the point), which is the clue for finding the angle.
\(\arctan\)	arctangent	The inverse of the tangent (also written \(\tan^{-1}\)). It returns the angle \(\theta\) with \(\tan\theta = k\). The angles it returns are limited to between \(-\dfrac{\pi}{2}\) and \(\dfrac{\pi}{2}\), so a quadrant adjustment is needed for angles in Quadrants II and III.
\(\pi\)	pi	The ratio of a circle's circumference to its diameter (about 3.14159). In radian measure, a half turn, \(180^{\circ}\), is exactly \(\pi\). The Greek letter \(\pi\) is said to come from the first letter of the Greek word for "perimeter".
\(n\)	en	A letter often used for an integer, from the first letter of "number". On this page it is how many full turns the angle is shifted, including negative integers (turning the other way) and 0 (no turn).

Terms

rectangular coordinates	A way to give the position of a point as a pair \(\left(x,\ y\right)\): how far across and how far up. These are the everyday coordinates used since middle school, also called Cartesian coordinates.
polar coordinates	A way to give the position of a point as a pair \(\left(r,\ \theta\right)\): the distance \(r\) from a reference point and the angle \(\theta\) from a reference direction. They are taught in Precalculus.
pole	The reference point from which distances are measured in polar coordinates. On the coordinate plane it is placed at the origin \(O\).
polar axis	The reference ray from which angles are measured in polar coordinates. It starts at the pole, and on the coordinate plane it lies along the positive \(x\)-axis.
radius	The distance \(r\) from the pole to the point \(P\) (also called the radial coordinate). Strictly, the segment (or ray) joining the pole and \(P\) is also meant, with length \(r\). It is 0 or greater.
angle θ	The angle \(\theta\) of the direction from the pole to the point \(P\), measured counterclockwise from the polar axis (also called the polar angle or argument). Adding a full turn (\(2\pi\)) gives the same direction, so answers are usually given in a range such as \(0 \le \theta < 2\pi\) or \(-\pi < \theta \le \pi\).
quadrant	One of the four regions the two axes divide the coordinate plane into. The upper right is Quadrant I, and counting counterclockwise from there come Quadrants II, III and IV. Knowing which quadrant a point is in is the clue for fixing the angle.
quadrant adjustment	Within one full turn, two angles satisfy \(\tan\theta = \dfrac{y}{x}\), so you choose the one that matches the quadrant of the point (for example, by adding \(\pi\) to the acute angle from \(\arctan\)). This fix is the quadrant adjustment.
unit circle	The circle of radius 1 centered at the origin. The point on it in the direction of the angle \(\theta\) is \(\left(\cos\theta,\ \sin\theta\right)\), which is the basis of the polar conversion formulas.
radian measure	Measuring an angle by the length of the arc on a circle of radius 1. \(180^{\circ} = \pi\) radians. From trigonometric functions in Algebra 2 and Precalculus on, radians are the standard unit rather than degrees.
special angles	\(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\) and the angles related to them (these plus multiples of \(90^{\circ}\)). Their trigonometric ratios can be written exactly, such as \(\dfrac{1}{2}\), \(\dfrac{\sqrt{2}}{2}\) and \(\dfrac{\sqrt{3}}{2}\), and most textbook and test problems use them.
polar equation	An equation of a curve written as a relationship between \(r\) and \(\theta\). For example, \(r = 2\) is the circle of radius 2 centered at the origin, and \(\theta = \dfrac{\pi}{4}\) is the ray from the origin with slope 1. Figures determined by the distance from a center, such as circles and spirals, have much simpler equations than in rectangular coordinates.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to review these topics is the fastest way forward.

The coordinate plane (Grades 5–6)	Being able to write the position of a point as a pair \(\left(x,\ y\right)\) Knowing where Quadrants I to IV are and the signs of \(x\) and \(y\) in each
Square roots and radicals (Grade 8 and Algebra 1)	Knowing that \(\sqrt{2}\) is the positive number whose square is 2 Being able to simplify a root by taking out square factors, as in \(\sqrt{8} = 2\sqrt{2}\)
The Pythagorean theorem (Grade 8)	Knowing that the hypotenuse of a right triangle with legs \(a\) and \(b\) is \(\sqrt{a^2 + b^2}\) Knowing that the distance from the origin to a point on the coordinate plane can be found with the Pythagorean theorem
Trigonometric ratios and the unit circle (Geometry, Algebra 2 and Precalculus)	Knowing that \(\sin\), \(\cos\) and \(\tan\) are the vertical position, horizontal position and slope of a point on the unit circle Knowing the trigonometric values of \(30^{\circ}\), \(45^{\circ}\) and \(60^{\circ}\) (\(\dfrac{1}{2}\), \(\dfrac{\sqrt{2}}{2}\), \(\dfrac{\sqrt{3}}{2}\)) Being able to find trigonometric ratios of angles over \(90^{\circ}\) with the unit circle (for example, \(\cos 135^{\circ} = -\dfrac{\sqrt{2}}{2}\))
Radian measure (Algebra 2 and Precalculus)	Being able to convert between degrees and radians with \(180^{\circ} = \pi\) radians (for example, \(60^{\circ} = \dfrac{\pi}{3}\))
Polar coordinates (Precalculus)	Knowing what the pole, the polar axis, the radius and the angle refer to Knowing that the same point has more than one set of polar coordinates (shifting the angle by \(2\pi\) gives the same point)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table for rectangular → polar

x-coordinate	3
y-coordinate	4
Radius r	=SQRT(B1^2+B2^2)
Angle θ (radians, 0 ≤ θ < 2π)	=MOD(ATAN2(B1,B2),2*PI())
Angle θ (degrees)	=DEGREES(MOD(ATAN2(B1,B2),2*PI()))

Table for polar → rectangular

Radius r	2
Angle θ (degrees)	60
x-coordinate = r cosθ	=B1*COS(RADIANS(B2))
y-coordinate = r sinθ	=B1*SIN(RADIANS(B2))

Table for another angle for the same point

Angle θ (degrees)	60
Number of turns n (integer, may be negative)	1
Shifted angle (degrees)	=B1+360*B2

After pasting, the upper rows are your inputs and the lower rows are calculated automatically.
In the first table, ATAN2(x-coordinate, y-coordinate) returns the angle with the quadrant adjustment already done (more than −π and up to π). Taking MOD with 2π turns it into an angle from 0 up to (but not including) 2π. For the example point (3, 4), r = 5 and θ ≈ 0.9273 (about 53.13 degrees).
In the second table, RADIANS converts degrees to radians. For the example r = 2 and θ = 60 degrees, x = 1 and y ≈ 1.7320508 (= √3). Excel gives decimal answers, so use the calculator on this page when you want exact values such as √3 or fractions of π.
The third table makes other polar coordinates for the same point. Adding one full turn, 360 degrees, to the example 60 degrees gives 420 degrees. With n = −1 you get −300 degrees, and all of these are the same point.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table for rectangular → polar

x-coordinate	3
y-coordinate	4
Radius r	=SQRT(B1^2+B2^2)
Angle θ (radians, 0 ≤ θ < 2π)	=MOD(ATAN2(B1,B2),2*PI())
Angle θ (degrees)	=DEGREES(MOD(ATAN2(B1,B2),2*PI()))

Table for polar → rectangular

Radius r	2
Angle θ (degrees)	60
x-coordinate = r cosθ	=B1*COS(RADIANS(B2))
y-coordinate = r sinθ	=B1*SIN(RADIANS(B2))

Table for another angle for the same point

Angle θ (degrees)	60
Number of turns n (integer, may be negative)	1
Shifted angle (degrees)	=B1+360*B2

The same formulas as in Excel work as is (ATAN2 takes its arguments in the same order, "x-coordinate, y-coordinate"). Copy the whole table, paste it into cell A1, and replace the numbers with your own.

How to calculate it in Python

import math

# Rectangular → polar
x_coordinate = 3
y_coordinate = 4
radius = math.sqrt(x_coordinate ** 2 + y_coordinate ** 2)
argument = math.atan2(y_coordinate, x_coordinate)   # includes the quadrant adjustment (−π < θ ≤ π)
if argument < 0:
    argument += 2 * math.pi                         # move into the range 0 ≤ θ < 2π
print(f"Radius r = {radius}")
print(f"Angle θ = {argument} rad ({math.degrees(argument)} degrees)")

# Polar → rectangular
r = 2
theta_degree = 60
theta = math.radians(theta_degree)
print(f"x-coordinate = {r * math.cos(theta)}")
print(f"y-coordinate = {r * math.sin(theta)}")

math.atan2(y, x) returns the angle with the quadrant adjustment already done, so unlike arctan it gives the right direction for points in Quadrants II and III too. Its arguments are in the order "y, x", the opposite of how coordinates are written, which is an easy place to slip. The first half of this example converts the point (3, 4); running it shows r = 5.0 and θ ≈ 0.9272952180016122 rad (about 53.13 degrees). The second half is the reverse conversion with r = 2 and θ = 60°, giving x ≈ 1.0000000000000002 and y ≈ 1.7320508075688772 (= √3). x is not exactly 1 because of rounding error in decimal (floating-point) numbers; to get exact values you need to calculate with symbols, as the calculator on this page does. With the cmath module from the standard library, cmath.polar(complex(3, 4)) also gives the radius and the angle (−π < θ ≤ π) in one step.

How to write it in LaTeX and other math languages (copy and paste)

From polar to rectangular (two ways to give the location of a point)

x = r cosθ, y = r sinθ

x = r\cos\theta,\quad y = r\sin\theta

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mtable columnalign="left">
    <mtr><mtd>
      <mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mi>&#x3B8;</mi>
    </mtd></mtr>
    <mtr><mtd>
      <mi>y</mi><mo>=</mo><mi>r</mi><mi>sin</mi><mi>&#x3B8;</mi>
    </mtd></mtr>
  </mtable>
</math>

x = r cos theta, y = r sin theta

{x, y} = {r Cos[theta], r Sin[theta]}

x := r*cos(theta); y := r*sin(theta);

[x, y] = pol2cart(theta, r);

x = r cos θ, y = r sin θ

Radius \(r\) (distance from the pole)

r = √(x² + y²)

r = \sqrt{x^{2} + y^{2}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>r</mi>
    <mo>=</mo>
    <msqrt>
      <mrow>
        <msup><mi>x</mi><mn>2</mn></msup>
        <mo>+</mo>
        <msup><mi>y</mi><mn>2</mn></msup>
      </mrow>
    </msqrt>
  </mrow>
</math>

r = sqrt(x^2 + y^2)

r = Sqrt[x^2 + y^2]

r := sqrt(x^2 + y^2);

r = hypot(x, y);

r = √(x^2 + y^2)

Angle \(\theta\) (from tanθ and the quadrant)

tanθ = y/x

\tan\theta = \dfrac{y}{x}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>tan</mi><mi>&#x3B8;</mi>
    <mo>=</mo>
    <mfrac><mi>y</mi><mi>x</mi></mfrac>
  </mrow>
</math>

tan theta = y/x

theta = ArcTan[x, y]

theta := arctan(y, x);

theta = atan2(y, x);

tan θ = y/x

The same point has more than one set of polar coordinates

(r, θ) = (r, θ + 2nπ)

\left(r,\ \theta\right) = \left(r,\ \theta + 2n\pi\right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>(</mo><mi>r</mi><mo>,</mo><mi>&#x3B8;</mi><mo>)</mo>
    <mo>=</mo>
    <mo>(</mo><mi>r</mi><mo>,</mo>
    <mi>&#x3B8;</mi><mo>+</mo><mn>2</mn><mi>n</mi><mi>&#x3C0;</mi>
    <mo>)</mo>
  </mrow>
</math>

(r, theta) = (r, theta + 2 n pi)

theta + 2 n Pi

theta + 2*n*Pi;

theta2 = theta + 2*n*pi;

(r, θ) = (r, θ + 2nπ)

How to have ChatGPT do the calculation

You are a math assistant for polar coordinates in the plane. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

1. Convert the point (3, 4) in rectangular coordinates to polar coordinates (r, θ). Show the radius r and the angle θ (in the range 0 ≤ θ < 2π, in both radians and degrees).
2. Convert the point (2, 60°) in polar coordinates to rectangular coordinates (x, y). Show x and y as decimals and, if possible, as exact values with √.

In Python, use math.atan2 and math.sqrt, and show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula