Conversion result
Step-by-step work
Calculator
Rectangular and polar graphs
Implicit graphs use numerical contour sampling. Explicit polar forms use sampled angle values.
Calculation history
| Date | Mode | Input | Result | |
|---|---|---|---|---|
| No calculations yet. | ||||
Formula used
The calculator substitutes x = r cos(θ) and y = r sin(θ). It also uses x² + y² = r² and tan(θ) = y/x. These identities preserve the same geometric curve while changing the coordinate system.
x = r cos(θ) y = r sin(θ) r² = x² + y² tan(θ) = y/x
How to use this calculator
- Select the required conversion mode.
- Enter a valid equation, point, parametric pair, or batch list.
- Choose radians or degrees and set decimal precision.
- Select whether to solve for r, preserve implicit form, or verify the result.
- Press Convert equation, then inspect the steps and graphs.
- Copy, print, share, or export the calculation when finished.
Example conversions
| Type | Rectangular form | Polar form |
|---|---|---|
| Rectangular equation | x^2 + y^2 = 25 | r = 5 |
| Vertical line | x = 4 | r = 4 / cos(θ) |
| Horizontal line | y = 3 | r = 3 / sin(θ) |
| Line through origin | y = 2x | tan(θ) = 2 |
| Circle through pole | x^2 + y^2 = 8x | r = 8cos(θ) |
| Circle through pole | x^2 + y^2 = 6y | r = 6sin(θ) |
| Diagonal line | x + y = 10 | r = 10 / (cos(θ) + sin(θ)) |
| Hyperbola | x^2 - y^2 = 6 | r^2 cos(2θ) = 6 |
Understanding rectangular and polar equations
Rectangular coordinates describe every point using horizontal and vertical distances. Polar coordinates describe the same point using a radius and an angle. Equation conversion replaces Cartesian variables without changing the represented curve.
The radius measures distance from the pole, which is the polar origin. The angle measures rotation from the positive horizontal axis. Negative radii represent points facing in the opposite angular direction.
Circles centered at the origin become especially simple in polar form. A rectangular circle such as x² plus y² equals twenty-five becomes r equals five. Lines and shifted circles usually retain trigonometric terms after conversion.
Some equations can be solved explicitly for r, while others remain implicit. An implicit answer is still valid when r and theta appear together. Domain restrictions matter when division introduces zeros in a denominator.
The graph comparison helps confirm that both equations trace the same shape. Numerical contour plotting is useful for complicated implicit equations. Symbolic verification additionally reverses the substitutions and compares equivalent forms.
Exact mode keeps roots, fractions, pi, and trigonometric expressions when possible. Decimal mode is useful for points and numerical checks. Precision controls affect displayed approximations rather than the underlying symbolic conversion.
Frequently asked questions
1. What is the main rectangular-to-polar substitution?
Replace x with r cos(θ) and y with r sin(θ). Then simplify with x² + y² = r².
2. Must every answer be solved for r?
No. Many correct polar equations remain implicit because isolating r is impossible or unhelpful.
3. Can the calculator convert straight lines?
Yes. It handles vertical, horizontal, diagonal, and general linear equations.
4. Does it support conic sections?
Yes. Circles, ellipses, parabolas, and hyperbolas can be substituted and simplified.
5. What does a negative radius mean?
A negative radius points in the direction opposite the stated angle.
6. Why can a denominator restriction appear?
Solving for r may require division by a trigonometric expression that can equal zero.
7. Can I convert points instead of equations?
Yes. Point mode calculates radius with the distance formula and angle with atan2.
8. How does batch mode work?
Enter one rectangular equation per line. The calculator converts every valid line independently.
9. Are the graphs exact?
Explicit curves are sampled densely. Implicit curves use numerical contours, so graphing remains an approximation.