2.5
4.330127
5
60
°
1.047198
rad
2.5
4.330127
5
60
°
1.047198
rad
The Polar Equation Calculator converts between polar coordinates $$(r, \theta)$$ and Cartesian coordinates $$(x, y)$$. Polar coordinates describe a point by its distance from the origin and the angle from the positive x-axis, offering a natural framework for problems involving circular symmetry, spirals, and angular motion.
In polar form, the conversion relationships are $$x = r\cos\theta$$ and $$y = r\sin\theta$$. Conversely, $$r = \sqrt{x^2 + y^2}$$ and $$\theta = \arctan(y/x)$$. These transformations are foundational in physics, engineering, and advanced mathematics, appearing in everything from antenna radiation patterns to complex analysis.
The calculator implements the standard polar-Cartesian conversion formulas derived from trigonometric definitions on the unit circle.
Polar to Cartesian: Given $$(r, \theta)$$, the Cartesian coordinates are:
$$x = r \cos\theta$$
$$y = r \sin\theta$$
Cartesian to Polar: Given $$(x, y)$$, the polar coordinates are:
$$r = \sqrt{x^2 + y^2}$$
$$\theta = \arctan\!\left(\frac{y}{x}\right)$$
The calculator uses the two-argument arctangent function (atan2) to correctly determine the angle in all four quadrants, returning values in the range $$[0°, 360°)$$. Negative radii are supported, since a negative $$r$$ with angle $$\theta$$ is equivalent to positive $$|r|$$ at angle $$\theta + 180°$$.
Angles are accepted in degrees and internally converted to radians for computation using $$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$$. The radian equivalent is also displayed in the output for convenience.
The output shows both coordinate representations of the same point. When converting from polar to Cartesian, the $$(x, y)$$ values represent the horizontal and vertical displacement from the origin. When converting from Cartesian to polar, the radius $$r$$ gives the straight-line distance from the origin, while $$\theta$$ indicates the counterclockwise angle from the positive x-axis.
A result of $$\theta = 0°$$ means the point lies on the positive x-axis, $$90°$$ on the positive y-axis, $$180°$$ on the negative x-axis, and $$270°$$ on the negative y-axis. Points at the origin have $$r = 0$$ with an undefined angle.
Inputs
Results
x = 5 cos 60° = 5 × 0.5 = 2.5; y = 5 sin 60° = 5 × 0.8660 ≈ 4.3301.
Inputs
Results
r = √(9 + 16) = 5; θ = arctan(4/3) ≈ 53.13°. This is the classic 3-4-5 right triangle.
Polar coordinates are ideal when the problem has circular or radial symmetry. Examples include orbital mechanics, antenna patterns, spiral curves, and integrating over circular regions. Any equation of the form $$r = f(\theta)$$ is naturally expressed in polar form.
A negative radius $$r$$ means the point is reflected through the origin. The point $$(-r, \theta)$$ is equivalent to $$(|r|, \theta + 180°)$$. The calculator computes $$x = r\cos\theta$$ and $$y = r\sin\theta$$ directly, which automatically handles negative $$r$$.
When converting from Cartesian coordinates $$(0, 0)$$, the angle is mathematically undefined since $$\arctan(0/0)$$ is indeterminate. The calculator returns $$0°$$ by convention, but the angle has no geometric meaning at the origin.
Yes. Every point $$(x, y)$$ in the Cartesian plane can be represented in polar form, though the representation is not unique. The same point can be described as $$(r, \theta)$$, $$(r, \theta + 360°)$$, or $$(-r, \theta + 180°)$$.
A complex number $$z = x + iy$$ can be written in polar form as $$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$, where $$r = |z|$$ is the modulus and $$\theta = \arg(z)$$ is the argument. This is Euler's formula and is fundamental in signal processing and quantum mechanics.
The calculator uses IEEE 754 double-precision floating-point arithmetic, providing approximately 15-16 significant decimal digits of precision. Results are displayed to 6 decimal places, which is sufficient for virtually all practical applications.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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