- Polar area: A = 1/2 ∫[a→b] r(θ)² dθ
- Polar arc length: L = ∫[a→b] √( r(θ)² + (dr/dθ)² ) dθ
- Type your polar function using theta.
- Enter start and end angles for the interval.
- Select degrees or radians, then choose results.
- Set intervals and method for desired accuracy.
- Press Calculate to see results above the form.
| θ | r(θ) | Notes |
|---|---|---|
| 0 | 2 | Starts on positive x-axis. |
| π/2 | 3 | Maximum radius for this curve. |
| π | 2 | Symmetry point across y-axis. |
| 3π/2 | 1 | Minimum radius for this curve. |
| 2π | 2 | Completes one full rotation. |
Polar area and arc length overview
Polar curves are written as r(θ), where radius changes with angle. This calculator estimates area and arc length on a selected interval. It helps with roses, cardioids, spirals, and custom plots. Use it when a closed form is unavailable.
1) What the area result means
The reported area uses A = ½∫ r(θ)² dθ. The square keeps contributions non‑negative. For self‑intersecting curves, the value represents accumulated sector area over the interval you choose.
2) What the arc length result means
Arc length uses L = ∫ √(r(θ)² + (dr/dθ)²) dθ. It blends radius and how quickly radius changes. Rapid oscillations can increase length even when the average radius stays similar.
3) Numerical method and accuracy
The calculator performs numerical integration using Simpson’s rule (recommended) or the trapezoidal rule. Simpson’s rule often converges faster for smooth functions. Increase n for cusps, tight spirals, or terms like sin(20*theta). For arc length, dr/dθ is estimated by a central difference, so jagged functions may need higher n to avoid noise.
4) Angle units and interval design
You may enter limits in degrees or radians. Internally, everything is computed in radians, so degree inputs are converted automatically. Common choices are 0→2π for one rotation or 0→π for half‑rotation symmetry. If you accidentally reverse the limits, the calculator swaps them for you.
5) Function syntax you can use
Enter expressions with theta, constants pi and e, and functions like sqrt, abs, ln, and exp. Write explicit multiplication (2*sin(theta)). Powers use ^, for example (1+cos(theta))^2.
6) Where these results are used
Polar areas appear in antenna patterns, mechanical cams, and decorative layouts. Arc length supports material estimates for bent parts, toolpaths, and plotted designs. Exporting to CSV helps document inputs, while the PDF report is easy to share.
7) Practical tips for reliable outputs
If you see unexpected values, simplify the expression and test a smaller range first. Then increase n until results stabilize. If the function is undefined at a limit (for example, ln(theta) at theta=0), shift the range slightly. When a curve crosses the pole, try breaking the interval into pieces and comparing runs.
FAQs
1) Does the calculator require radians?
No. You can enter degrees or radians. Degree limits are converted to radians automatically, and all trig functions are evaluated in radians internally for consistency.
2) Why does arc length need a derivative?
Arc length depends on how the curve changes with angle. The term dr/dθ measures radial change, and combining it with r(θ) captures the true path length.
3) What value of n should I choose?
Start with 500–1500 for smooth curves. Use larger values for cusps, tight spirals, or rapid oscillations. Increase n until the result changes very little.
4) Why do I get an invalid result?
This usually happens when the expression is undefined at some θ (division by zero, log of a non‑positive number, or sqrt of a negative value). Adjust the formula or angle limits.
5) Can I use negative r(θ)?
Yes, but interpretation may vary. The area integral uses r(θ)², so contributions stay non‑negative. For self‑intersections, the computed area is accumulated over the chosen interval.
6) What do the exports include?
CSV and PDF exports include your inputs, the computed area and/or arc length, method details, and a small set of sample points. Recalculate to refresh the stored export data.