Polar Graphing Calculator
Type one or more polar equations r = f(θ) and watch them drawn on a polar grid. Drag the θ range sliders, tune the a and k parameters, hover to read points and download the graph as a PNG.
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Type θ as t, theta or the θ key. Examples: 3sin(2t), 1 - cos(t), t/pi, e^(0.1t). Use a and k to link a formula to the sliders.
Measurements over the θ range
| Curve | Area ½∫r² dθ | Arc length | Max |r| |
|---|
How to use the Polar Graphing Calculator
- Type a polar equation in the r₁ box, for example 3cos(2θ). Use t, theta or θ for the angle.
- Add more curves with + Add function, or pick one from the example list.
- Set the θ range with the Start and End sliders or a quick range button. Roses with fractional k and spirals need more than 2π.
- Explore: drag to pan, scroll or pinch to zoom, press ⤢ to fit, and hover a curve to read its coordinates.
- Press Draw animation to watch the curve being traced, then Download PNG to save the graph.
Common polar curves and their equations
| Curve | Equation | Shape notes | θ range to close |
|---|---|---|---|
| Circle through the origin | r = 2a cos θ or r = 2a sin θ | Circle of radius a centered on an axis | 0 to π |
| Circle at the origin | r = a | Radius a | 0 to 2π |
| Rose | r = a cos(kθ) | k petals if k is odd, 2k petals if k is even | π (odd k), 2π (even k) |
| Cardioid | r = a(1 + cos θ) | Heart shape with a cusp at the origin | 0 to 2π |
| Limaçon | r = b + a cos θ | Inner loop if |b| < |a|, dimple if |a| < |b| < 2|a| | 0 to 2π |
| Lemniscate of Bernoulli | r² = a² cos(2θ) | Figure eight; type sqrt(a^2 cos(2θ)) and the negative root is drawn by symmetry | 0 to 2π |
| Archimedean spiral | r = aθ | Turns are evenly spaced, 2πa apart | Open; longer range = more turns |
| Logarithmic spiral | r = a e^(bθ) | Same angle with every radius; grows by e^(2πb) per turn | Open |
| Line not through the origin | r = d sec(θ − φ) | Straight line at distance d from the origin | Between −π/2 and π/2 around φ |
Converting between polar and Cartesian coordinates
Every point on the graph has two names. From polar to Cartesian use x = r cos θ and y = r sin θ. Going back, r = √(x² + y²) and θ is the angle of the point, found with the two-argument arctangent atan2(y, x) so the quadrant comes out right. The hover readout shows both forms at once, which is a quick way to check homework conversions.
Symmetry tests save time when sketching by hand: replacing θ with −θ and getting the same equation means the curve is symmetric about the x-axis (polar axis); replacing θ with π − θ tests the y-axis; and replacing r with −r tests symmetry about the origin.
Tips for exploring polar graphs
- Link a formula to the sliders with
aandk:a cos(kθ)lets you drag through roses with 3, 4, 5 or fractional petals. - Implicit multiplication works:
3sin(2t),2θanda(1 + cos θ)are all understood. Use^for powers andpior π for π. - For the area enclosed by one petal, set the θ range to exactly one petal (for
cos(3θ), from −π/6 to π/6) before reading the area column. - Draw animation is the clearest way to see where r turns negative: watch the dot jump to the opposite side of the origin.
Frequently asked questions
What is a polar equation?
A polar equation gives the distance r from the origin as a function of the angle θ measured counterclockwise from the positive x-axis. Each pair (θ, r) is converted to the Cartesian point x = r cos θ, y = r sin θ, and joining those points draws the curve.
How do I graph a rose curve?
Type r = a cos(kθ) or a sin(kθ). If k is an odd whole number the rose has k petals and is complete after θ goes from 0 to π; if k is even it has 2k petals and needs 0 to 2π. For a fraction k = p/q in lowest terms, extend the range (for example 0 to 4π for k = 2.5) to close the curve.
What happens when r is negative?
A negative r is plotted in the opposite direction, at angle θ + π. That is why r = cos(2θ) still draws four petals and why the limaçon r = 1 + 2 sin θ has an inner loop. The calculator follows this standard convention automatically.
How is the area of a polar curve calculated?
The area swept as θ runs from α to β is A = ½∫ r² dθ. The tool evaluates this integral numerically over your θ range. If the range traces part of a curve twice, that part is counted twice, so r = cos θ from 0 to 2π gives π/2 even though the circle encloses π/4; use 0 to π for the single loop.
How is the arc length found?
Arc length in polar form is L = ∫ √(r² + (dr/dθ)²) dθ. The calculator samples thousands of points along the curve and adds the lengths of the tiny straight pieces between them, which matches the integral to several decimal places for smooth curves.
Can I use degrees instead of radians?
Yes. Choose Degrees in the angle unit menu. The θ range is then shown in degrees, θ in your formula is in degrees and sin, cos and tan take degrees, so r = 2 + 2cos(θ) looks the same in both modes.
Why is part of my curve missing?
The curve is skipped wherever the formula has no real value, for example sqrt(cos(2θ)) when cos(2θ) is negative, or at an asymptote such as sec θ at θ = π/2. Check the θ range too: many curves need more than one turn to close.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.