Polar Coordinates Calculator

Two ways to name the same point

A street address and a GPS reading describe the same house. Rectangular and polar coordinates do the same job for points on a plane. The rectangular form says how far to go sideways (x) and how far up (y). The polar form says how far to walk in a straight line (r) and at what angle off the horizontal (theta). Both names point at the identical spot, and converting between them is two lines of trigonometry.

The formulas, both directions: x = r x cos(theta) and y = r x sin(theta) go from polar to rectangular, while r = sqrt(x^2 + y^2) and theta = atan2(y, x) go back. The r in polar coordinates is a plain distance from the origin, never a signed position, though the notation does allow a negative r with a special meaning covered below. NASA Glenn Research Center keeps a short introduction to rectangular and polar coordinates that builds the same two formulas for describing aircraft motion, and the NIST Digital Library of Mathematical Functions lists the sine and cosine values every conversion runs on.

Rectangular to polar: r is easy, the angle needs care

The distance r comes from the Pythagorean theorem: r = sqrt(x^2 + y^2). The point (3, 4) gives r = sqrt(9 + 16) = sqrt(25) = 5, and triangles like this 3-4-5 one are the classic textbook cases because the arithmetic lands on a whole number.

The angle is where mistakes happen. The plain arctangent of y / x cannot tell quadrants apart: the ratio 4 / 3 = 1.3333 whether the point is (3, 4) in the first quadrant or (-3, -4) in the third, yet those two points sit 180 degrees apart. The atan2 function fixes this by taking y and x separately and returning the angle in the correct quadrant, between -180 and 180 degrees. For (3, 4), atan2(4, 3) = 53.13 degrees. Add 360 degrees if you prefer angles from 0 to 360.

Polar to rectangular: one multiplication each

Going the other way needs no quadrant logic at all, just two multiplications: x = r x cos(theta) and y = r x sin(theta). A point at r = 6 and theta = 120 degrees has x = 6 x cos(120 degrees) = 6 x (-0.5) = -3 and y = 6 x sin(120 degrees) = 6 x 0.8660 = 5.196. The point lands in the second quadrant, x negative and y positive, exactly where an angle of 120 degrees should put it.

Negative r does not break the rule; it means walk backward along the angle direction. A point at r = -2 and theta = 60 degrees sits at x = -2 x cos(60 degrees) = -1 and y = -2 x sin(60 degrees) = -1.732, the same place as r = 2 and theta = 240 degrees. Two alias rules cover everything: (r, theta) and (-r, theta + 180 degrees) name the same point, and so do (r, theta) and (r, theta + 360 degrees). A full turn returns you to the start, which is why one point carries endless polar names: (5, 53.13 degrees), (5, 413.13 degrees) and (-5, 233.13 degrees) all sit on the same dot.

Degrees and radians, and picking one

Angles come in two currencies. Degrees divide the circle into 360 parts. Radians measure the arc itself: 180 degrees equals pi radians, about 3.14159, and 1 radian is 57.2958 degrees. Calculators and programming languages usually expect radians, while everyday work usually speaks degrees, so most conversion mistakes trace back to this switch left in the wrong position.

Converting is one multiplication. Degrees to radians: multiply by pi / 180. Radians to degrees: multiply by 180 / pi. The table shows the angles worth memorizing, with the cosine and sine values the conversions run on. NIST treats the radian as the coherent SI unit of plane angle, with the degree allowed alongside it, which is why instrument readouts offer both (see the NIST Guide for the Use of the International System of Units).

The calculator answers in both units at once, so the output row that reads 53.13 degrees also reads 0.9273 radians, and you can quote whichever your audience expects.

Worked examples, forward and back

Start with (3, 4). The radius: r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. The angle: atan2(4, 3) = 53.13 degrees. So the polar form is (5, 53.13 degrees). Now check it by converting back: x = 5 x cos(53.13 degrees) = 5 x 0.6 = 3 and y = 5 x sin(53.13 degrees) = 5 x 0.8 = 4. The round trip lands exactly where it started, which is the quickest way to catch a slip.

Two patterns worth noticing in the table. Points on an axis have angles that snap to 0, 90, 180 or 270 degrees, because one coordinate is zero. And a point and its mirror across the x axis share the same r, with angles that are negatives of each other.

Polar equations and the shapes they draw

Coordinates name points; polar equations draw curves. An equation like r = 3 says every point sits 3 units out, which is a circle of radius 3. An equation like theta = 45 degrees says the angle never changes, which is a straight line through the origin. More interesting shapes appear when r depends on theta, and a few of them are famous enough to have names.

The third row shows why polar form earns its keep: a circle that passes through the origin and bulges along the x axis is one short equation in polar form and a clumsy one in rectangular form. The cardioid, the curve traced by a point on a rolling circle, is one line in polar and a mess outside it. When a problem has a natural center and a natural direction, polar coordinates usually shorten the algebra.

Bearings, radar and other everyday uses

Polar thinking is everywhere once you look for it. Radar screens are polar by design: the operator reads a distance out and an angle around. Navigation bearings, though, run on the mirror convention: bearing 0 is north and bearings increase clockwise, while the math angle theta = 0 points east and increases counterclockwise. Converting between them takes one step: bearing = 90 degrees - theta, wrapped into 0 to 360. The math angle of 53.13 degrees from the worked example is a bearing of 90 - 53.13 = 36.87 degrees, which points north northeast.

The same pair of numbers runs telescope mounts, robot arms, CNC machines and the camera systems that follow objects across a room. Speaker and antenna engineers publish polar plots that show loudness or signal strength at each angle around the device. Game developers store positions in rectangular form for physics but reach for polar form for anything that orbits, spins or fans out. The common thread: whenever the natural question is how far and which way, polar coordinates are the vocabulary that fits.

Common mistakes and quick checks

Degrees where radians belong. Feed cos(53.13) into a calculator set to radians and you get -0.9619 instead of 0.6, an error that hides well because every output still looks plausible. Decide the unit first, then convert.

atan2 arguments in the wrong order. The convention is atan2(y, x), and swapping them answers a different question: atan2(4, 3) is 53.13 degrees, but atan2(3, 4) is 36.87 degrees, the bearing of the same point. Both are useful; only one is theta.

Forgetting the quadrant. A point in quadrant III reported with a first quadrant angle is off by exactly 180 degrees. When x is negative, plain arctan lies; use atan2 or add 180 degrees by hand.

Confusing r with x. The radius is the hypotenuse, never the horizontal leg, so r is always at least as large as |x| and |y|. A conversion that returns r smaller than one of the coordinates has gone wrong somewhere.

Redundant forms are not errors. An output of theta = -53.13 degrees and one of theta = 306.87 degrees describe the same direction; so do r = 5 and r = -5 with the angle flipped 180 degrees. If two tools disagree by exactly 360 degrees, or by the flip rule, both are right.