Calculator
Formula used
The calculator uses the standard rectangular to polar identities.
x = r cos(θ)y = r sin(θ)x² + y² = r²It also changes x² into r²cos²(θ), y² into r²sin²(θ), and xy into r²sin(θ)cos(θ).
How to use this calculator
- Enter a rectangular equation using x and y.
- Use the caret symbol for powers, such as x^2.
- Select the theta display style.
- Add an optional point if you need r and theta.
- Press Calculate to view the polar equation.
- Use CSV or PDF buttons to save the result.
Example data table
| Rectangular equation | Polar substitution result | Useful note |
|---|---|---|
| x^2 + y^2 = 25 | r^2 = 25 | Circle with radius 5. |
| x = 4 | r cos(θ) = 4 | Vertical line in rectangular form. |
| y = 3 | r sin(θ) = 3 | Horizontal line in rectangular form. |
| x^2 + y^2 = 2x | r^2 = 2r cos(θ) | Can simplify to r = 2cos(θ). |
| x^2 - y^2 = 9 | r^2cos^2(θ) - r^2sin^2(θ) = 9 | Use trig identities if needed. |
Why Convert X Y Equations to Polar Form?
Rectangular equations use x and y to describe a curve. Polar equations use r and theta. Both forms describe the same plane. A conversion helps when a curve is circular, radial, or angle based. Many circles, spirals, and conic sections look cleaner in polar notation.
How the conversion works
This calculator replaces each rectangular variable with its polar identity. It uses x = r cos(theta) and y = r sin(theta). It also applies x squared plus y squared equals r squared. The result shows the new equation and the main substitutions. You can compare the original form with the polar form quickly.
Where polar form helps
Polar form is useful in calculus and geometry. Area, arc length, and rotational motion often depend on angles. A curve that looks hard in x and y may become simple with r and theta. For example, x squared plus y squared equals 25 becomes r squared equals 25. That gives r equals 5. This is a circle centered at the origin.
Reading the output
Some equations need algebra after substitution. The calculator gives the converted structure first. You may still divide, factor, square, or isolate r. Use the precision option for numeric point conversion. Enter an x value and a y value to see the matching radius and angle.
Learning and export use
Teachers can use this page for examples. Students can use it to check homework steps. Designers and engineers can use it for radial layouts. The CSV export helps save table results. The PDF export gives a printable summary.
Accuracy tips
Always check the domain of theta. Also check whether negative r values are allowed. Polar coordinates can represent the same point in more than one way. A point can use r and theta, or negative r with theta plus pi. This flexibility is useful, but it can confuse beginners.
Use exact values when possible. Keep square roots and trigonometric terms during symbolic work. Round only at the final stage. This keeps answers accurate. A clean polar equation should show the same curve as the original equation. For best results, type powers with the caret symbol. Use x^2 and y^2. Keep multiplication signs between grouped terms. Review displayed substitutions before using the answer in a final solution.
FAQs
1. What does this calculator convert?
It converts rectangular equations written with x and y into polar form using r and theta substitutions.
2. Which formulas are used?
It uses x = r cos(theta), y = r sin(theta), and x squared plus y squared equals r squared.
3. Can it isolate r automatically?
It gives the main polar substitution. Some equations still need factoring, division, or trigonometric identities to isolate r.
4. What input style should I use?
Use simple algebra notation. Write powers as x^2 and y^2. Add an equal sign for complete equations.
5. Does it convert points too?
Yes. Enter optional x and y point values. The calculator returns radius and angle in degrees and radians.
6. Why is polar form useful?
Polar form is useful for circular curves, radial motion, spirals, and problems where angles are important.
7. Can I download results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable result summary.
8. Can one point have many polar forms?
Yes. Polar coordinates can repeat because angles wrap around. Negative radius values can also describe the same point.