Polar / Cartesian Calculator

r,θ — Point Conversion

(r,θ)↔(x,y) with quadrant identification

A — Polar Area

A = ½∫r²dθ via Gaussian quadrature

L — Arc Length

L = ∫√(r²+(r')²)dθ, dr/dθ numerical

r↔xy — Equation Convert

Pattern-match & substitution rules

d — Distance

Polar distance formula + midpoint

e·d — Conic in Polar

r=ed/(1±e·cos/sinθ), classify by e

dy/dx — Slope

(r'sinθ+rcosθ)/(r'cosθ−rsinθ)

↔ — Symmetry

Polar axis, θ=π/2, pole tests + curve ID

Polar → Cart \(x=r\cos\theta,\;y=r\sin\theta\)

Cart → Polar \(r=\sqrt{x^2+y^2},\;\theta=\operatorname{atan2}(y,x)\)

Area \(A=\frac{1}{2}\int_{\alpha}^{\beta}r^2\,d\theta\)

Arc Length \(L=\int_{\alpha}^{\beta}\sqrt{r^2+(r'(\theta))^2}\,d\theta\)

Distance \(d=\sqrt{r_1^2+r_2^2-2r_1r_2\cos\Delta\theta}\)

Conic \(r=\frac{ed}{1\pm e\cos\theta}\)

Slope \(\frac{dy}{dx}=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}\)

Sym: x-axis \(r(\theta)=r(-\theta)\)

Sym: y-axis \(r(\theta)=r(\pi-\theta)\)

Sym: pole \(r(\theta+\pi)=r(\theta)\)

Circle (origin) \(r=a\)

Circle (x-axis) \(r=2a\cos\theta\)

Circle (y-axis) \(r=2a\sin\theta\)

Cardioid \(r=a(1\pm\cos\theta)\)

Limaçon \(r=a\pm b\cos\theta\)

Rose (n petals) \(r=a\cos(n\theta)\)

Archimedean \(r=a\theta\)

Ellipse (e<1) \(r=\frac{ed}{1-e\cos\theta}\)

Parabola (e=1) \(r=\frac{d}{1+\cos\theta}\)

Hyperbola (e>1) \(r=\frac{ed}{1+e\cos\theta}\)

⊙Polar / Cartesian Calculator