Polar Graphing Calculator
Plot polar curves from equations using angle intervals, visualize shapes accurately, and compute points automatically with smooth, interactive graphics rendering.
Inputs
Equation Preview
Helping Notes
Enter the curve as r(t). Use radians for angles. Negative r is handled by rotating the angle by π automatically.
Angles like pi, 2*pi, and pi/3 are accepted. Sampling is automatic for a smooth graph without extra settings.
Results
Inputs Summary
Polar Plot
Computation Details
What Is a Polar Graphing Calculator?
A Polar Graphing Calculator lets you visualize and analyze curves defined in polar form . Instead of the Cartesian relation , polar graphs use a radius measured from the origin at an angle from the positive x‑axis. The calculator draws the curve over a chosen angle range, finds intercepts at the pole, marks special points (max/min radius), and computes geometric quantities like tangent direction, enclosed area, and arc length. It is ideal for precalculus, calculus, physics, computer graphics, and engineering design whenever spiral or petal‑like shapes occur naturally.
About the Polar Graphing Calculator
The tool converts polar to Cartesian for plotting via . Slope is obtained by differentiating : wherever the denominator is nonzero. The calculator highlights symmetries using quick tests (e.g., replace or ). It computes area swept between angles by , and arc length by . For curves given implicitly (e.g., ), the tool treats the valid angle sub‑ranges where and joins lobes appropriately.
Polar → Cartesian:
Polar slope:
Area:
Arc length:
Pole intercepts:
How to Use This Polar Graphing Calculator
- Enter (use radians). Set the angle interval or multiples of for full plots.
- Optionally enable derivatives to show slope and tangents, and toggle symmetry tests to simplify the plotting range.
- Click calculate to draw the graph, locate pole intercepts, and compute and over the chosen interval.
- Inspect step‑by‑step formulas and copy exact or numerical results into notes or assignments.
Examples
- Cardioid: → symmetric about the x‑axis; area over is .
- Rose (four petals): ; one petal on ; total length via .
- Archimedean spiral: (for )—use partial intervals to limit coil count; compute sector area between angles.
- Circle of radius R: across any ; length , area .
Formula Snippets Ready for Rendering
FAQs
How do I choose the angle range?
Use a fundamental interval that traces the whole curve—often or smaller if symmetry repeats petals.
Can I plot negative r values?
Yes. Negative reflects the point across the origin by radians.
Why are radians recommended?
Polar calculus formulas assume radians; convert degrees via .
How do I find tangents?
Use the polar slope formula. Vertical tangents occur where the denominator vanishes while the numerator doesn’t.
Can the calculator find enclosed area automatically?
Yes—provide . For multi‑petal roses, sum area over one petal and multiply by petal count.
What about self‑intersections?
Polar curves often retrace points. The graph shows overlaps; lengths and areas use your chosen angle bounds.
Does it handle implicit relations like r² = a² cos 2θ?
Yes, by restricting to angles with real and plotting each lobe’s interval separately.