Analyze loops, ellipses, and custom parametric shapes. Tune intervals, density, scaling, and orientation controls instantly. Plot curves, verify closure, and export polished calculation reports.
Use t as the parameter. Powers may use ^. Bounds accept forms like 2*pi.
The filled region uses the same closure logic as the numeric area calculation.
Example curve: x(t) = 4cos(t) and y(t) = 2sin(t) for 0 ≤ t ≤ 2π.
| Parameter t | x(t) | y(t) |
|---|---|---|
| 0 | 4.0000 | 0.0000 |
| π/4 | 2.8284 | 1.4142 |
| π/2 | 0.0000 | 2.0000 |
| 3π/4 | -2.8284 | 1.4142 |
| π | -4.0000 | 0.0000 |
| 5π/4 | -2.8284 | -1.4142 |
| 3π/2 | 0.0000 | -2.0000 |
| 7π/4 | 2.8284 | -1.4142 |
For a closed parametric curve x = x(t) and y = y(t), the signed area is A = 1/2 ∫(x(t)y′(t) − y(t)x′(t)) dt over the chosen parameter interval.
This calculator samples the curve, then applies the polygon shoelace form A ≈ 1/2 Σ[xiyi+1 − xi+1yi]. That makes the result practical for custom expressions, scaled curves, and shifted loops.
Positive signed area usually means counterclockwise travel. Negative signed area usually means clockwise travel. Absolute area ignores direction and reports pure enclosed size.
Closed loops work best because enclosed area is well-defined. Open paths can still be processed when auto-close is enabled, but the last point connects back to the first point before area is measured.
Signed area keeps orientation information. Counterclockwise tracing often produces a positive value, while clockwise tracing usually produces a negative value. Absolute area removes that sign and keeps only magnitude.
The calculator estimates area from sampled points. More samples usually mean a smoother boundary and better numerical stability, especially for sharp turns, petals, cusps, or rapidly changing oscillations.
A closure gap means the first and last sampled points do not match closely. If auto-close is enabled, the calculator adds a final segment. If disabled, you should adjust the interval.
Yes. You may use expressions such as sin(t), cos(3*t), sqrt(abs(t)), and powers with ^. Bounds also accept constants like 2*pi.
Pure shifting does not change the enclosed area of a closed curve. Scaling does change it. If you double horizontal scale and vertical scale, area increases by four times.
The centroid estimates the geometric center of the enclosed region. It becomes unreliable when the area is nearly zero, self-cancels, or the sampled path is numerically degenerate.
Use CSV when you want the complete sampled dataset for spreadsheets or audits. Use PDF when you need a quick summary report containing equations, settings, metrics, and a compact preview table.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.