Polar Coordinates Converter & Calculator
Convert between polar (r, θ) and rectangular (x, y) coordinates, and calculate quantities in both systems.
Polar Coordinates Guide
Polar Coordinate System
In polar coordinates, a point is specified by: r — distance from the origin (pole). θ — angle from the positive x-axis (counterclockwise positive). Conversion to Cartesian: x = r cos θ. y = r sin θ. Conversion from Cartesian: r = √(x² + y²). θ = arctan(y/x) — with quadrant correction. Quadrant check: if x > 0: θ as calculated. If x < 0: add 180° (or π). If x = 0, y > 0: θ = 90°. If x = 0, y < 0: θ = 270°. Use atan2(y, x) function in calculators and programming for automatic correct quadrant.
When Polar Is Easier Than Cartesian
Polar excels for problems with: rotational symmetry. Curves naturally described by angle and distance. Examples: circle of radius a centred at origin: r = a (much simpler than x² + y² = a²). Spiral: r = aθ (Archimedean spiral) or r = ae^(kθ) (logarithmic spiral). Cardioid: r = a(1 + cos θ) — a heart shape. Rose curves: r = a cos(nθ) — n petals if n odd, 2n petals if n even. Limaçon: r = a + b cos θ — varies depending on a/b ratio. Lemniscate: r² = a² cos(2θ) — figure of 8. Cartesian equations fo
Distance and Area in Polar
Distance between two polar points (r₁, θ₁) and (r₂, θ₂): d = √(r₁² + r₂² − 2r₁r₂ cos(θ₁ − θ₂)). Derived from the cosine rule. Area enclosed by polar curve from θ=α to θ=β: A = ½ ∫[α,β] r² dθ. Example: area of circle r = 2: A = ½ ∫[0,2π] 4 dθ = ½ × 4 × 2π = 4π. ✓ Same as πr² with r=2. Area between two curves: A = ½ ∫[α,β] (r₂² − r₁²) dθ. Polar coordinate calculus is essential for analysing planetary orbits, electromagnetic field patterns, antenna radiation patterns.
Applications
Navigation and surveying: polar coordinates correspond directly to bearing and distance. GPS systems convert internally between coordinate systems. Astronomy: positions on the celestial sphere use spherical coordinates (3D polar). Engineering: radar systems naturally output polar data. Robotics: arm rotation often uses polar/cylindrical coordinates. Mathematics: contour integration in complex analysis uses polar form extensively. Engineering: antenna patterns (radiation pattern of an aerial) are
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