Polar ↔ Rectangular Coordinate Converter (Free)
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A polar rectangular converter switches a point between (x, y) and (r, θ). To go from rectangular to polar, use r = √(x² + y²) and θ = atan2(y, x): the point (−3, 4) becomes (5, 126.87°). To go back, use x = r cos θ and y = r sin θ: (4, 30°) becomes (2√3, 2) ≈ (3.464, 2).
The point (−3, 4) in polar coordinates is (r, θ) ≈ (5, 126.8699°), or θ ≈ 2.2143 in radians.
Polar grid: circles every 2 units, rays every 30°.
Show steps
- Distance from the origin: r = √(x² + y²) = √((−3)² + 4²) = √(9 + 16) = √25 = 5.
- The point is in Quadrant II (x is negative, y is positive). Reference angle: α = arctan(|y| ÷ |x|) = arctan(4 ÷ 3) ≈ 53.1301°.
- Place the angle in the right quadrant: θ = 180° − α ≈ 126.8699°. This is exactly what atan2(y, x) does.
- Why not just arctan(y/x)? A calculator returns arctan(−1.3333) ≈ −53.1301°, which points into the opposite quadrant. Because x is negative, you must add 180°.
- Polar coordinates: (r, θ) ≈ (5, 126.8699°). Check: r cos θ ≈ −3.0000 = x and r sin θ ≈ 4.0000 = y.
Tip: r is never negative here. Adding 360° (or 2π) to θ, or using −r with θ + 180°, names the same point.
How to use the polar ↔ rectangular converter
- Choose a direction: Rectangular → polar, Polar → rectangular, or Equations to convert a whole equation such as r = 4 cos θ.
- Pick the angle unit, degrees or radians. In radians you can type π: π/3, 2pi/3 or −3π/4.
- Enter the numbers. Fractions and roots work too, such as −1/2, 2√3 or sqrt(2)/2, so special-angle answers come out exact.
- Read the converted point (exact form and decimal), the quadrant, the reference angle and the other names for the same point. The polar grid shows the point, its radius and its angle, and Show steps explains every line.
Polar and rectangular coordinates
Rectangular (Cartesian) coordinates (x, y) say how far to move right and up from the origin. Polar coordinates (r, θ) say how far to go, r, and in which direction, θ, measured counterclockwise from the positive x-axis. Both describe the same plane, so every point has both kinds of address. Polar coordinates suit circles, spirals and anything that turns around a center; rectangular coordinates suit lines and grids.
Polar to rectangular conversion formulas
| Direction | Formulas |
|---|---|
| Polar → rectangular | x = r cos θ, y = r sin θ |
| Rectangular → polar | r = √(x² + y²), θ = atan2(y, x) |
| Always true | x² + y² = r², tan θ = y/x (when x ≠ 0) |
The formulas come from a right triangle with hypotenuse r: the horizontal leg is r cos θ and the vertical leg is r sin θ. The Pythagorean theorem gives r.
Finding θ in the correct quadrant
The trap in rectangular-to-polar conversion is the angle. A calculator’s arctan(y/x) only returns angles between −90° and 90°, so it cannot tell (3, −4) from (−3, 4). Work out the reference angle α = arctan(|y| ÷ |x|), then place it using the signs of x and y. This is exactly what the two-argument function atan2(y, x) does.
| Signs of (x, y) | Quadrant | θ (degrees) | θ (radians) |
|---|---|---|---|
| (+, +) | I | α | α |
| (−, +) | II | 180° − α | π − α |
| (−, −) | III | 180° + α | π + α |
| (+, −) | IV | 360° − α | 2π − α |
Points on an axis need no arctan: (5, 0) has θ = 0°, (0, 5) has 90°, (−5, 0) has 180° and (0, −5) has 270°.
Exact values for special angles
| θ | cos θ | sin θ |
|---|---|---|
| 0° (0) | 1 | 0 |
| 30° (π/6) | √3/2 | 1/2 |
| 45° (π/4) | √2/2 | √2/2 |
| 60° (π/3) | 1/2 | √3/2 |
| 90° (π/2) | 0 | 1 |
For other multiples of 30° and 45°, use the same values with the signs of the quadrant. The converter recognizes these angles in both directions and gives exact answers such as 3√3 instead of 5.1962.
Worked examples
Example 1: (−3, 4) to polar
r = √((−3)² + 4²) = √25 = 5. The reference angle is arctan(4/3) ≈ 53.13°. The point is in Quadrant II, so θ = 180° − 53.13° = 126.87°. The answer is (5, 126.87°), or about (5, 2.2143) in radians. Plain arctan(4/(−3)) gives −53.13°, which points into Quadrant IV.
Example 2: (6, π/3) to rectangular
x = 6 cos(π/3) = 6 × 1/2 = 3 and y = 6 sin(π/3) = 6 × √3/2 = 3√3 ≈ 5.196. The point is (3, 3√3). Set the unit to radians first; 6 cos(π/3 degrees) would be a very different number.
Example 3: (−1, −√3) to polar in radians
r = √(1 + 3) = 2. The ratio |y|/|x| = √3 gives a 60° (π/3) reference angle, and both coordinates are negative, so θ = π + π/3 = 4π/3. In the range −π to π the same point is (2, −2π/3).
Example 4: a negative radius
(−2, 135°) means “face 135°, then walk 2 units backward”. x = −2 cos 135° = −2 × (−√2/2) = √2 and y = −2 sin 135° = −√2, so the point is (√2, −√2) in Quadrant IV, the same as (2, 315°).
Converting polar and rectangular equations
To turn a rectangular equation into polar form, substitute x = r cos θ, y = r sin θ and x² + y² = r², then solve for r. To go the other way, rearrange so that r appears as r cos θ, r sin θ or r²; multiplying both sides by r often helps.
| Polar | Rectangular | Graph |
|---|---|---|
| r = a | x² + y² = a² | circle centered at the origin |
| θ = c | y = (tan c)x | line through the origin |
| r = a cos θ | x² + y² = ax | circle through the origin, center (a/2, 0) |
| r = a sin θ | x² + y² = ay | circle through the origin, center (0, a/2) |
| r = a sec θ | x = a | vertical line |
| r = a csc θ | y = a | horizontal line |
Example: r = 4 cos θ. Multiply by r: r² = 4r cos θ, so x² + y² = 4x. Completing the square gives (x − 2)² + y² = 4, a circle with center (2, 0) and radius 2. The Equations tab does this for circles and lines, in either direction, and draws the graph.
Polar form of complex numbers
The same conversion writes a complex number a + bi in polar form r(cos θ + i sin θ), with r = |a + bi| and θ = arg(a + bi). For −3 + 4i, r = 5 and θ ≈ 126.87°. Polar form makes multiplying and taking powers of complex numbers much easier.
Common mistakes with polar coordinates
- Trusting arctan(y/x) alone. It is off by 180° whenever x is negative.
- Mixing degrees and radians. cos 60 in radian mode is about −0.95, not 0.5.
- Forgetting that r can be negative. (−r, θ) is the same point as (r, θ + 180°).
- Thinking a polar point has one name. (5, 126.87°), (5, −233.13°) and (−5, 306.87°) are all the same point.
- Rounding special angles. Keep 2√3 exact until the last step when the problem asks for exact answers.
Checking homework with the converter
Do the conversion by hand first: sketch the point, decide the quadrant and estimate the angle. Then enter the same numbers here. If your answer differs, open Show steps and compare each line, starting with the quadrant and the angle unit. Finally, convert your answer back the other way to see that you return to the starting point.
Polar and rectangular coordinates practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
r = √(36 + 64) = 10 and θ = arctan(8/6) ≈ 53.13° in Quadrant I. 14 adds the coordinates instead of using the Pythagorean theorem; 36.87° swaps x and y.
x = 8 cos 60° = 8 × 1/2 = 4 and y = 8 sin 60° = 8 × √3/2 = 4√3. Swapping cos and sin gives (4√3, 4).
The reference angle is 45°, and both coordinates are negative (Quadrant III), so θ = 180° + 45° = 225°. arctan(−5/−5) = 45° is the wrong quadrant.
Using −r turns the direction around by 180°: (−3, 20°) points to 200°. (3, −200°) is the same as (3, 160°).
cos(3π/2) = 0 and sin(3π/2) = −1, so x = 0 and y = −2: the point is on the negative y-axis.
r = √((−2)² + (2√3)²) = √(4 + 12) = √16 = 4. 16 forgets the square root.
Multiply by r: r² = 6r sin θ, so x² + y² = 6y, a circle with center (0, 3) and radius 3.
Replace x with r cos θ: r cos θ = 4, so r = 4/cos θ = 4 sec θ. r = 4 cos θ is a circle.
The point is in Quadrant II, but arctan returns a Quadrant IV angle. Add 180°: −53.13° + 180° = 126.87°.
x² + y² = r², so r² = 25 and r = 5: a circle of radius 5 centered at the origin.
Quiz complete
Frequently asked questions
How do you convert rectangular to polar?
Use r = √(x² + y²) and θ = atan2(y, x). For (−3, 4), r = 5 and θ = 126.87°, so the polar form is (5, 126.87°). A plain arctangent gives −53.13°, which must be moved into Quadrant II by adding 180°.
What are the polar rectangular conversion formulas?
Polar to rectangular: x = r cos θ and y = r sin θ. Rectangular to polar: r = √(x² + y²) and θ = atan2(y, x). Keep your calculator in the same angle unit as the problem.
How do you convert polar to rectangular coordinates?
Multiply r by the cosine and sine of θ. For (4, 30°), x = 4 cos 30° = 2√3 ≈ 3.464 and y = 4 sin 30° = 2, so the point is (2√3, 2).
How do you convert a polar equation to rectangular form?
Replace r cos θ with x, r sin θ with y and r² with x² + y². For r = 4 cos θ, multiply by r to get r² = 4r cos θ, which is x² + y² = 4x, a circle with center (2, 0) and radius 2.
Is polar to Cartesian the same as polar to rectangular?
Yes. Cartesian and rectangular coordinates are two names for the same (x, y) system. For example, (2, 90°) becomes (0, 2) in both.
How do you convert polar to rectangular with radians?
Use the same formulas with θ in radians. For (6, π/3), x = 6 cos(π/3) = 3 and y = 6 sin(π/3) = 3√3 ≈ 5.196. Select radians first, or the answer will be wrong.
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