Calculator
Formula used
For a region bounded by an outer curve r = ro(θ) and an inner curve r = ri(θ), the area in polar coordinates is:
The centroid coordinates are estimated with:
ȳ = (1/A) · (1/3) ∫ ( ro(θ)3 − ri(θ)3 ) sin(θ) dθ
This tool estimates integrals numerically (Simpson or Trapezoid). Increase segments for smoother curves or sharp turns.
How to use this calculator
- Choose angle bounds θ₁ and θ₂, then select degrees or radians for those bounds.
- Pick presets for outer and inner curves, or enter custom equations using t.
- Use radians inside equations. If bounds are degrees, they are converted automatically.
- Select Simpson for better accuracy and set segments (try 800–5000 for most shapes).
- Click Calculate. Use the download buttons to export results.
Example data table
| Outer r(t) | Inner r(t) | Bounds | Expected area |
|---|---|---|---|
| 5 | 0 | 0 → 2π | 25π ≈ 78.5398 |
| 2*(1+cos(t)) | 0 | 0 → 2π | 6π ≈ 18.8496 |
| 4*cos(t) | 2*cos(t) | −π/2 → π/2 | 3π ≈ 9.4248 |
Polar region area in one pass
This calculator estimates the area of a bounded region described in polar coordinates. You enter an outer curve r(t), an optional inner curve, and an angle interval. The output is the signed area, plus a centroid estimate, so you can move from sketches to numbers without manual tables.
Why polar form matters
Many shapes are simplest around the pole: circles r=a, cardioids r=a(1+cos t), limacons r=a+b cos t, roses r=a cos(nt), and spirals r=a+bt. In polar form, the boundary is one function of angle, so the area integral is compact and avoids converting to x–y first.
Setting theta limits
Correct bounds are the biggest accuracy lever. Use 0 to 2π for full rotations, or smaller spans for petals and arcs. If you know symmetry, integrate one symmetric slice, then multiply outside the tool. Degree bounds are converted to radians internally, but the equations still interpret trig inputs as radians.
Building r(t) curves
Pick a preset to reduce typos, then adjust a, b, n, and the trig choice. For custom work, type equations using t, pi, and e with operators + − * / and ^. Supported functions include sin, cos, tan, sqrt, abs, ln, log, and exp. Keep curves real-valued across the interval.
Accuracy and segment choices
The integral is computed numerically using Simpson’s rule or the trapezoid method. Simpson generally converges faster on smooth curves, but needs an even segment count. Start around 800 segments for typical classroom problems, then raise to 5000+ when your curve has sharp cusps or rapid oscillation.
Reading centroid results
Along with area, the tool estimates x̄ and ȳ using cubic-radius moments. It also reports the centroid in polar form (r̄, θ̄). If the reported area is near zero, centroid values can become unstable, which usually indicates your bounds cancel or the region is extremely thin.
Exporting and reusing outputs
After computing, download CSV to archive inputs and results in spreadsheets, or generate a lightweight PDF report for sharing. For repeat studies, keep the same bounds and change only one parameter, such as a or n, to see how area scales. This supports quick sensitivity checks and design comparisons. Use exports to document assumptions and verify peer reviews.
FAQs
What does the “inner curve” mean?
The inner curve represents a hole or excluded region, like an annulus. The area is computed from the difference ½∫(ro²−ri²)dθ. Set inner to zero when the region starts at the pole.
Why can the area become negative?
If your θ interval runs backward, or if the inner and outer curves swap order for part of the range, the integrand can be negative. Enable “Absolute integrand” to report a positive magnitude when crossings are expected.
Should I use degrees or radians?
Use either for the bounds selector. However, the equation box always evaluates trig in radians. If you prefer degrees, convert inside the equation by using t*pi/180, or keep bounds in degrees and equations in radians.
How many segments should I choose?
For smooth curves, 800–2000 segments is usually sufficient. Increase to 5000–20000 for cusps, high-frequency roses, or tight spirals. If results change noticeably when doubling segments, keep increasing until the change is small.
What equations are supported in custom mode?
You can use numbers, t, pi, e, parentheses, and operators + − * / ^. Supported functions are sin, cos, tan, asin, acos, atan, sqrt, abs, ln, log, and exp. Avoid functions that make r(t) undefined within bounds.
Why does the centroid look unstable?
Centroid formulas divide by area. If the region’s net area is near zero due to cancellation, the centroid can blow up. Try adjusting bounds to isolate one region, disable crossings, or verify that your curves describe a single, filled shape.