Choose an Equation Template
Use the matching template, then enter its coefficients. Unused fields are ignored automatically.
Example Data Table
| Surface | Rectangular form | Cylindrical form | Main replacement |
|---|---|---|---|
| Cylinder | x² + y² = 25 | r² = 25 | x² + y² → r² |
| Sphere | x² + y² + z² = 49 | r² + z² = 49 | x² + y² → r² |
| Plane | 2x − 3y + z = 6 | r(2cos θ − 3sin θ) + z = 6 | x, y → r cos θ, r sin θ |
| Paraboloid | x² + y² = 4z | r² = 4z | x² + y² → r² |
Formula Used
Cylindrical coordinates replace the horizontal rectangular coordinates with a radius and angle. Use these standard relationships:
- x = r cos θ
- y = r sin θ
- x² + y² = r²
- z = z
Use r² whenever x² and y² appear as a pair. Use cosine and sine substitutions for individual x or y terms.
How to Use This Calculator
- Select the equation pattern that matches your rectangular equation.
- Enter the radius, scale, coefficients, or constant needed by that pattern.
- Choose the manual guide for an equation outside the listed templates.
- Select Convert Equation to place the result above the form.
- Review the source, cylindrical result, and substitution formula.
- Download a CSV or PDF copy when you need saved work.
Coordinate Conversion Guide
Why cylindrical form helps
Rectangular equations describe points with x, y, and z coordinates. Cylindrical equations use r, theta, and z instead. Both systems locate the same point. The difference is the way horizontal position is expressed. Rectangular form separates horizontal movement into two perpendicular directions. Cylindrical form combines that movement into distance from the z-axis and an angle around it. This makes circular surfaces easier to describe and visualize.
Core substitutions
The most important relationship is r squared equals x squared plus y squared. It removes separate x and y squared terms. Also replace x with r cosine theta. Replace y with r sine theta. Leave z unchanged. These substitutions convert many common equations. A cylinder becomes very simple. A sphere keeps z and radial distance. A cone becomes a relationship between r and z.
A reliable conversion method
Start by identifying patterns inside the rectangular equation. Look for x squared plus y squared first. Treat that pair as r squared. Then inspect remaining x or y terms. Replace each one with its trigonometric cylindrical expression. Keep parentheses around cosine and sine terms. They prevent coefficient mistakes. Finally, simplify both sides without changing the meaning. Write theta when an angle remains. Do not introduce theta when all terms became r squared.
Common surfaces
A vertical circular cylinder begins as x squared plus y squared equals R squared. Its cylindrical form is r squared equals R squared. This may also be written as r equals R. A sphere begins as x squared plus y squared plus z squared equals R squared. Replace the horizontal pair, producing r squared plus z squared equals R squared. The new equation shows why cylindrical coordinates suit rotationally symmetric surfaces.
Linear terms and coefficients
Planes can require more care. For a x plus b y equals d, substitute each linear term. The converted expression becomes r times a cosine theta plus b sine theta equals d. A z coefficient stays unchanged. Thus, a x plus b y plus c z equals d becomes r times a cosine theta plus b sine theta plus c z equals d. The angle remains because direction matters for a nonradial plane.
Checks before using a result
Use consistent units for every coefficient and constant. Confirm exponents before submitting values. A missing square can change a cylinder into another surface. Check whether a radius or scale factor should be positive. Review the displayed source equation before trusting a result. The calculator uses standard substitutions and template rules. For unusual equations, use the manual guide shown by the custom option. It explains the substitutions, but it cannot simplify every possible expression automatically.
Where the method is useful
Conversion is useful beyond homework. Engineers model pipes, tanks, and rotational components. Scientists describe fields around central axes. Designers create surfaces with circular symmetry. Students also use cylindrical form in triple integrals. The correct coordinate system can shorten setup and calculation steps. Practice equation families. Compare the original and converted surfaces. Soon, the substitutions will become familiar and reliable.
Frequently Asked Questions
1. What changes when converting rectangular equations?
Replace x with r cos θ, y with r sin θ, and x² + y² with r². Keep z exactly as it is.
2. Why does x² + y² become r²?
In cylindrical coordinates, r measures distance from the z-axis. The Pythagorean relationship gives r² = x² + y².
3. Does every equation contain theta?
No. Equations containing only x² + y² often convert using r² alone. Theta appears when separate x or y terms remain.
4. How is a cylinder written in cylindrical form?
A cylinder x² + y² = R² becomes r² = R². For a nonnegative radius, it can also be written as r = R.
5. How is a sphere converted?
A sphere x² + y² + z² = R² becomes r² + z² = R². The z coordinate does not change.
6. Why do planes use sine and cosine?
Individual x and y terms require their direct substitutions. This preserves each plane coefficient and the plane direction around the z-axis.
7. Can this convert every possible equation automatically?
The listed templates calculate common forms. The manual option provides standard substitutions for unusual equations that need hand simplification.
8. Should radius values be positive?
Yes. Radius represents distance and is nonnegative. A negative value normally indicates the same magnitude with a changed angular interpretation.
9. What units should I enter?
Use one consistent unit system. Coefficients and constants must match the units used by the original equation.
10. Can I use the result in triple integrals?
Yes. Cylindrical equations often simplify circular regions. Remember that volume integrals also include the Jacobian factor r.
11. How can I improve conversion accuracy?
Practice strengthens conversion skills across many coordinate geometry problems.