About the Polar Graphing Calculator
This polar graphing calculator draws polar curves r = f(θ) — cardioids, roses, limaçons, spirals and lemniscates — and parametric curves x = x(t), y = y(t) such as ellipses, Lissajous figures and cycloids. Type the equation, set the parameter range (expressions like 2pi are fine) and the graph redraws instantly on a polar grid you can zoom.
Besides the picture it does the calculus: the area swept out, ½∫r² dθ for polar curves or the enclosed area ½|∮(x dy − y dx)| for parametric ones, the arc length of the curve, the farthest distance from the origin, and the exact point at any parameter value you choose. That makes it useful for calculus II homework on polar area and arc length, for precalculus students learning how the shape changes with the equation, and for teachers projecting examples.
Angles are in radians. You can write θ as theta, θ or t. Area and length are computed numerically from 4,000 samples, so they are typically accurate to 6 or more significant figures.
With the default inputs, the area (polar: ½∫r² dθ; parametric: net enclosed) is 4.712389. Change any value above to recalculate instantly.
How to use the polar graphing calculator
- 1Choose polar r = f(θ) or parametric x(t), y(t).
- 2Type the equation, using theta (or θ, t) as the angle in radians.
- 3Set the parameter range, e.g. 0 to 2pi.
- 4Watch the graph update and zoom if needed.
- 5Read the area, arc length and the point at your chosen θ.
Formula and method
A polar curve gives the distance r from the origin as a function of the angle θ; converting each point with x = r cos θ and y = r sin θ plots it on ordinary axes. The area swept by the radius from θ = α to β is ½∫r² dθ, because each thin wedge is approximately a circular sector of area ½r² dθ. Arc length adds up the lengths of tiny pieces of the curve.
For parametric curves the enclosed area of a closed loop follows from Green's theorem, A = ½|∮(x dy − y dx)|, and the arc length is ∫√(x′² + y′²) dt. This tool samples the curve at 4,000 points: the polar area uses Simpson's rule, the parametric area uses the shoelace formula, and the arc length sums the straight segments between samples.
- r, θ
- Polar radius and angle (radians)
- x(t), y(t)
- Parametric coordinates
- A
- Area
- L
- Arc length
Worked examples
Cardioid r = 1 + cos θ
A = ½∫₀^{2π}(1 + cos θ)² dθ = ½(2π + π) = 3π/2 ≈ 4.7124, and the cardioid's length is exactly 8. At θ = π/2, r = 1, which is the point (0, 1).
Four-petal rose r = cos 2θ
Over 0 to 2π the rose traces four petals once each. A = ½∫cos²(2θ) dθ = π/2 ≈ 1.5708, and the total arc length is about 9.6884.
Archimedean spiral r = θ
The area swept is ½∫₀^{2π} θ² dθ = (2π)³/6 ≈ 41.3417, and the arc length is ½[θ√(1 + θ²) + sinh⁻¹θ] at 2π ≈ 21.2563. At θ = π the spiral is at (−π, 0).
Parametric ellipse x = 3 cos t, y = 2 sin t
The enclosed area of an ellipse is πab = 6π ≈ 18.8496, and its perimeter (an elliptic integral with no elementary closed form) is about 15.8654. At t = 0 the point is (3, 0).
Frequently asked questions
How do you graph a polar equation?+
Pick values of θ, compute r for each, and plot the point at distance r from the origin in direction θ — equivalently at (r cos θ, r sin θ). Joining the points as θ increases traces the curve. This calculator does that with thousands of points.
What range of θ should I use?+
0 to 2π traces most curves once. Roses r = cos(kθ) with odd k repeat after π, so use 0 to π for those; spirals keep growing, so use a longer range to see more turns.
How do you find the area of a polar curve?+
Use A = ½∫r² dθ between the angles that trace the region once. For a cardioid r = 1 + cos θ from 0 to 2π this gives 3π/2. For regions between two curves, subtract ½∫r_inner² from ½∫r_outer².
What happens when r is negative?+
A negative r plots the point in the opposite direction, at angle θ + π. That is how a rose like r = cos 2θ draws petals in all four quadrants.
Can I plot parametric equations too?+
Yes. Switch to parametric mode and enter x(t) and y(t), e.g. x = cos 3t, y = sin 2t for a Lissajous curve. The calculator reports the enclosed area of closed curves and the arc length.