Calculator
Example data table
| Cartesian equation | Main substitution | Polar result |
|---|---|---|
| x^2 + y^2 = 25 | x^2 + y^2 = r^2 | r = 5 |
| x = 4 | x = r cos(θ) | r = 4 / cos(θ) |
| y = 3 | y = r sin(θ) | r = 3 / sin(θ) |
| y = 2x | tan(θ) = y / x | θ = arctan(2) |
| x^2 + y^2 - 6x = 0 | r^2 - 6r cos(θ) = 0 | r = 6 cos(θ) |
Formula used
The calculator uses the standard coordinate identities:
x = r cos(θ)
y = r sin(θ)
x^2 + y^2 = r^2
For a straight line Ax + By + C = 0, substitution gives r(A cos(θ) + B sin(θ)) + C = 0.
So the solved polar form is r = -C / (A cos(θ) + B sin(θ)).
For x^2 + y^2 + Dx + Ey + F = 0, the polar form becomes r^2 + Dr cos(θ) + Er sin(θ) + F = 0.
How to use this calculator
- Enter a Cartesian equation using x and y.
- Use ^ for powers, such as x^2.
- Select the angle symbol and decimal precision.
- Choose the theta sample range for the table.
- Press Calculate to view the polar form.
- Use CSV or PDF to save the result.
About Cartesian to polar conversion
A Cartesian equation uses x and y to describe a curve. A polar equation uses r and an angle. Both systems can describe the same point. The difference is the way the point is measured. Cartesian coordinates measure horizontal and vertical distance. Polar coordinates measure distance from the origin and direction from the positive x-axis.
This calculator changes common Cartesian forms into polar form. It is useful for circles, straight lines, and many second degree equations. It also shows the direct substitution. That step is important because it explains the algebra. You can see where every part of the polar equation comes from.
The main identities are simple. Replace x with r cos theta. Replace y with r sin theta. Replace x squared plus y squared with r squared. After substitution, collect like terms. Then solve for r when the equation allows it. Some equations produce one clean answer. Others produce a quadratic in r. In those cases, the calculator shows the polar relation and a solved formula.
Polar form is common in graphing, trigonometry, calculus, and analytic geometry. Circles centered at the origin become very simple. Lines through the origin often become angle equations. Circles shifted away from the origin often become equations like r equals a multiple of sine or cosine. These forms can make graph behavior easier to understand.
The sample table helps check the conversion. It calculates radius values for selected angles. It also converts those values back into x and y. This makes the result easier to verify. If no real radius exists for an angle, the table marks that case. This is helpful when the curve only appears over part of the angle range.
Use simple algebraic input for best results. Write multiplication clearly when needed. Use decimal or fractional coefficients. Keep all variables as x and y. The tool supports many common school and college patterns, but it is still wise to review each step. Coordinate conversion is easier when the original equation is clean.
FAQs
What does this calculator convert?
It converts Cartesian equations using x and y into polar equations using r and an angle symbol.
Which formulas are used?
It uses x = r cos(θ), y = r sin(θ), and x^2 + y^2 = r^2.
Can it solve every equation?
No. It supports many common polynomial forms. Complex symbolic expressions may need manual simplification after direct substitution.
Why is x^2 + y^2 replaced by r^2?
Because r is the distance from the origin. By the distance formula, r^2 equals x^2 plus y^2.
What does positive radius mean?
It keeps radius values greater than or equal to zero when sample values are generated.
Can I download the result?
Yes. Use the CSV button for spreadsheet data or the PDF button for a printable report.
What input format works best?
Use forms like x^2+y^2=25, x=4, y=2x+3, or x^2+y^2-6x=0.
Why are sample values included?
They help verify the polar equation by showing r values and matching Cartesian checks.