Formula Used
For one polar curve, the enclosed area is:
A = 1/2 ∫ r(θ)2 dθ
For the area between two polar curves, the calculator uses:
A = 1/2 ∫ |r1(θ)2 - r2(θ)2| dθ
If signed area is needed, uncheck the absolute area option. The final area also applies the symmetry multiplier and scale squared.
How to Use This Calculator
Enter the polar function using theta or t as the variable. Use operators such as +, -, *, /, and ^. Common functions include sin, cos, tan, sqrt, abs, log, and exp.
Choose single curve mode for one polar graph. Choose two curve mode when comparing inner and outer curves. Enter lower and upper angle limits. Then choose radians or degrees.
Select Simpson rule for smoother curves. Use more intervals for better accuracy. Press calculate to show the result above the form. Use the CSV or PDF buttons to save the result.
Example Data Table
| Curve |
Limits |
Mode |
Suggested method |
| 2*sin(3*theta) |
0 to pi |
Single curve |
Simpson rule |
| 1+cos(theta), 1 |
0 to 2*pi |
Between curves |
Simpson rule |
| 3*cos(theta) |
-pi/2 to pi/2 |
Single curve |
Trapezoid rule |
Understanding Area in Polar Curves
Polar curves describe points by distance and angle. The distance is called the radius. The angle is measured from a fixed ray. This form is useful for roses, spirals, cardioids, limacons, circles, and many symmetry based shapes.
Why Polar Area Needs a Special Formula
Rectangular area often uses vertical strips. Polar area uses thin circular sectors. Each small sector has area close to one half times radius squared times a tiny angle change. Adding all sectors gives the integral formula. This is why the square of the radius appears in the calculation.
Numerical Integration
Many polar curves do not produce a simple antiderivative. Some also have repeated petals or inner loops. This calculator uses numerical integration to handle such cases. Simpson rule is usually best for smooth functions. Trapezoid rule is simple and stable. Midpoint rule is helpful when endpoint behavior is less smooth.
Working With Two Curves
Area between two polar curves compares two squared radii. The larger region is found by subtracting one squared radius from the other. The absolute option is useful when the curves switch positions inside the interval. Signed area can still be useful for advanced checking.
Accuracy Tips
Use radians when possible, because most calculus formulas use radians. Increase intervals for curves with many waves, sharp turns, or narrow petals. Check the sample points to see how the radius changes. If symmetry is known, integrate one repeated part and apply the multiplier. This reduces work and improves review. Always choose limits that cover the intended loop or full region. A rose curve may need only one petal interval, while a cardioid often needs zero to two pi. The scale factor changes length units. Since area is squared, the final area multiplies by scale squared.
FAQs
What is a polar curve area?
It is the area enclosed by a curve written as r equals a function of theta. The calculator estimates that area over your chosen angle interval.
Can I enter theta as t?
Yes. You can use theta or t. Both represent the angle variable. You can also use pi in angle limits and expressions.
Which method should I choose?
Simpson rule is a good default for smooth polar curves. Trapezoid and midpoint rules are also available for comparison and special checks.
What does the symmetry multiplier do?
It multiplies the integrated area. Use it when you calculate one repeated petal, loop, or section and want the total repeated area.
How do I calculate area between two curves?
Select the two curve mode. Enter both polar functions and the shared angle limits. Keep absolute area checked when curves cross or switch order.
Why does area use r squared?
A small polar slice behaves like a circular sector. Sector area is one half times radius squared times angle width.
Can this calculator handle degrees?
Yes. Select degrees from the angle unit menu. The calculator converts degree limits internally before evaluating the calculus formula.
Why should I increase intervals?
More intervals usually improve numerical accuracy. This matters for curves with petals, oscillations, loops, or fast radius changes.