Convert between polar coordinates (r, θ) and rectangular coordinates (x, y). See the conversion formulas, step-by-step arithmetic, and a live canvas diagram showing the point on overlapping polar and Cartesian axes.
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Examples
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Conversion Formulas (SOH-CAH-TOA + Pythagorean Theorem)
x = r · cos θ y = r · sin θ
r = √(x² + y²) θ = atan2(y, x)
Draw a right triangle from the origin to the point (x, y). The hypotenuse is r, the horizontal leg is x, and the vertical leg is y.
By SOH-CAH-TOA: cos θ = x/r → x = r cos θ; sin θ = y/r → y = r sin θ.
By the Pythagorean theorem: r² = x² + y², so r = √(x² + y²).
A negative radius r is valid — it means the point is in the direction opposite to θ, i.e. (r, θ) and (−r, θ+π) name the same point.
Multiple Representations & atan2 vs arctan
(r, θ) = (r, θ+2π) = (−r, θ+π)
Every point in the plane has infinitely many polar representations because adding 2π to θ (one full revolution) lands on the same point. Changing the sign of r and adding π to θ also gives the same point.
atan2(y, x) is preferred over arctan(y/x) because it accounts for the quadrant automatically: arctan alone can't distinguish (1, 1) from (−1, −1) since both give arctan(1) = 45°. atan2 returns an angle in (−π, π] and this calculator maps it to [0, 2π) for the standard range.
The quadrant of (x, y) tells you which "slice" of the plane the angle falls in: Q I: 0–90°, Q II: 90–180°, Q III: 180–270°, Q IV: 270–360°.
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