Polar to Rectangular Coordinate Converter
Converts coordinates between polar form (r, θ) and rectangular form (x, y). Use it in engineering, physics, or mathematics whenever you need to switch between distance-and-angle and horizontal-vertical representations.
About this calculator
Polar coordinates represent a point by its radial distance r from the origin and an angle θ measured from the positive x-axis. Rectangular (Cartesian) coordinates represent the same point by its horizontal distance x and vertical distance y. The conversion formulas are: x = r · cos(θ) and y = r · sin(θ). Going the other direction: r = √(x² + y²) and θ = atan2(y, x). When θ is given in degrees it must first be converted to radians by multiplying by π/180 before applying the cosine or sine function. Angles in gradians are converted via multiplication by π/200. These identities come directly from the definition of sine and cosine in a right triangle formed by r, x, and y. Polar form is particularly convenient for describing circular motion, complex numbers, and electromagnetic fields, while rectangular form is easier for addition and subtraction of vectors.
How to use
Convert the polar point (r = 5, θ = 53.13°) to rectangular coordinates. Step 1 — find x: x = r · cos(θ) = 5 · cos(53.13° × π/180) = 5 · cos(0.9273 rad) = 5 × 0.6000 = 3.00. Step 2 — find y: y = r · sin(θ) = 5 · sin(53.13° × π/180) = 5 · sin(0.9273 rad) = 5 × 0.8000 = 4.00. So the polar point (5, 53.13°) corresponds to the rectangular point (3, 4). You can verify this using the Pythagorean theorem: √(3² + 4²) = √25 = 5, which matches the original radius r.
Frequently asked questions
When should I use polar coordinates instead of rectangular coordinates?
Polar coordinates are most convenient when a problem has circular or rotational symmetry. For example, describing the position of a satellite orbiting Earth, the path of a spiral, or the phase of an alternating-current signal is much simpler in polar form. Rectangular coordinates are better for translational problems, such as finding where two straight roads intersect or adding two force vectors. Many engineering fields—particularly RF engineering, control systems, and fluid dynamics—routinely switch between both systems depending on which simplifies the equations.
What is the difference between polar coordinates and spherical coordinates?
Polar coordinates are two-dimensional, describing a point in a plane using a radius r and an angle θ. Spherical coordinates extend this to three dimensions by adding a second angle: typically a radial distance ρ, a polar angle φ (measured from the z-axis), and an azimuthal angle θ (measured in the xy-plane). Spherical coordinates are used in 3-D physics problems such as gravitational fields, atomic orbital shapes, and GPS positioning. This calculator handles the two-dimensional polar-to-rectangular conversion only.
Why do I need to convert degrees to radians before computing sine and cosine?
The mathematical definitions of sine and cosine—and therefore every formula derived from them—assume angles measured in radians. Radians are the natural unit because they directly relate arc length to radius (arc = r·θ). Most calculators and programming languages use radians internally, so passing a degree value directly to sin() or cos() produces wrong answers. The conversion is simple: radians = degrees × π/180. For example, 90° becomes π/2 ≈ 1.5708 rad, and sin(π/2) = 1 as expected.