Calculator Inputs
Use the controls below to plot a polar equation and inspect numeric outputs.
Polar graphing formulas
Primary polar relation: r = f(θ)
Cartesian conversion: x = r cos(θ), y = r sin(θ)
Approximate enclosed area: A = 1/2 ∫ r2 dθ
Approximate arc length: L = ∫ √(r2 + (dr/dθ)2) dθ
The calculator samples many angle values between your chosen start and end limits. For each angle, it evaluates your equation, converts the polar point into Cartesian coordinates, and sends the complete set to the graph and table.
Area and arc length are numerical approximations. Their accuracy improves when you use a larger number of points, especially for shapes with sharp turns or rapid oscillation.
Step-by-step usage
- Enter a polar equation such as a * sin(b * theta) or choose a preset.
- Select degrees or radians for your angle input range.
- Set the start angle, end angle, and number of sampled points.
- Enter values for a, b, c, and d when your equation uses parameters.
- Choose marker and fill options if you want a different graph style.
- Press Submit to show the result above the form.
- Review the chart, summary metrics, and computed coordinate table.
- Use the export buttons to download CSV or PDF output.
Sample equation and values
Example equation: r = 2 sin(3θ) with display values shown in degrees.
| θ (degrees) | θ (radians) | r | x = r cos(θ) | y = r sin(θ) |
|---|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 30 | 0.5236 | 2.0000 | 1.7321 | 1.0000 |
| 60 | 1.0472 | 0.0000 | 0.0000 | 0.0000 |
| 90 | 1.5708 | -2.0000 | 0.0000 | -2.0000 |
| 120 | 2.0944 | 0.0000 | 0.0000 | 0.0000 |
FAQs
1) What is a polar graph?
A polar graph represents each point with a radius and an angle instead of x and y directly. It is useful for roses, spirals, cardioids, directional patterns, and other symmetry-heavy curves.
2) Which variable should I use in the equation?
Use theta for the angle. You can also use parameters a, b, c, and d to build reusable equations and quickly test different curve shapes.
3) Can I enter angles in degrees?
Yes. Choose degree mode and enter your range in degrees. The calculator converts those values internally so the trigonometric functions still evaluate correctly during plotting.
4) Why does a negative radius still produce valid points?
A negative radius points in the opposite direction of the given angle. The calculator converts that case into an equivalent positive-radius point for smooth plotting and consistent Cartesian coordinates.
5) How do I get a smoother curve?
Increase the number of points. More samples reduce visible gaps and improve the area and arc length estimates, especially when the equation changes rapidly over small angle intervals.
6) What functions can I use in the equation?
Common functions such as sin, cos, tan, sqrt, abs, log, exp, and powers work well. You can also use constants like pi and e inside the expression field.
7) What does the export feature include?
The CSV export includes computed point data. The PDF export includes summary metrics, a chart image, and a compact table of sampled points for easy sharing or printing.
8) Are the area and arc length values exact?
They are numerical approximations based on the chosen sample count. For most educational and practical uses, the estimates are very useful, and increasing points usually improves accuracy.