Polar Coordinates to Rectangular Equation Calculator

Calculator Inputs

Conversion Mode

Equation Type

a Value

b Value

c Value

r Value for Point Mode

Theta Value

Angle Unit

Decimal Precision

Horizontal Symbol

Vertical Symbol

Step Display

Show detailed steps

Custom Equation

Formula Used

The calculator uses these standard relationships between polar and rectangular coordinates:

x = r cos(θ)
y = r sin(θ)
r² = x² + y²
tan(θ) = y / x
cos(θ) = x / r
sin(θ) = y / r
sec(θ) = r / x
csc(θ) = r / y

The selected equation type decides which identity is applied first. Some equations need multiplication by r. Others need radical substitution or squaring.

How to Use This Calculator

Choose whether you want to convert a polar equation or a polar point.
Select the equation type closest to your problem.
Enter values for a, b, c, r, and theta where needed.
Select degrees or radians for angle-based entries.
Set the decimal precision for numeric answers.
Press the convert button to show the result above the form.
Use the CSV or PDF buttons to save the answer.

Example Data Table

Polar Form	Rectangular Form	Curve Type
r = 5	x² + y² = 25	Circle centered at origin
θ = 45°	y = x	Line through origin
r = 6 cos(θ)	x² + y² = 6x	Horizontal shifted circle
r = -4 sin(θ)	x² + y² = -4y	Vertical shifted circle
r = 3 sec(θ)	x = 3	Vertical line
r² = 16 sin(2θ)	(x² + y²)² = 32xy	Rose curve
r = 6 / (2 + 3 cos(θ))	2√(x² + y²) + 3x = 6	Conic style form

Understanding the Conversion

Polar equations use distance and angle. Rectangular equations use horizontal and vertical coordinates. This calculator connects both views. It replaces r, sine, cosine, tangent, secant, and cosecant with x and y relationships. The result helps you graph curves on ordinary coordinate planes.

Why Rectangular Form Matters

Many graphing systems prefer x and y. Teachers also ask for rectangular equations because they show centers, lines, circles, conics, and symmetry more clearly. A polar rule can look compact, yet the same curve may be easier to compare in rectangular form. For example, r equals a cosine theta becomes a shifted circle. Theta equals a constant becomes a straight line through the origin.

What the Tool Handles

The calculator supports points and common polar equations. You can convert constant radius equations, constant angle equations, sine circles, cosine circles, secant lines, cosecant lines, rose style equations, conic style equations, and limacon style equations. It also shows a substitution-only mode for custom entries. That mode is useful when the equation needs manual algebra after replacing the main polar identities.

Reading the Result

Each answer includes the main rectangular equation and a short explanation. When squaring is needed, the calculator also warns about possible restrictions. Squaring can introduce extra points if the original sign condition is ignored. Always compare the derived equation with the original polar rule when exact curve matching matters.

Good Input Habits

Use real numeric values. Choose degrees when your angle is written like 30 or 135. Choose radians when it is written using decimal radian measure. Set precision high for teaching solutions. Set it lower for quick graph labels. Use the x and y symbol boxes when your page uses custom variable names.

Practical Uses

This converter is useful for precalculus, calculus, analytic geometry, physics plotting, and engineering diagrams. It can support lesson examples, homework checking, and curve identification. The CSV export keeps records for tables. The PDF export creates a simple printable summary. Together, the steps, formulas, and examples make the conversion easier to audit. Because every result remains visible above the form, you can adjust inputs quickly. This saves time during practice. It also helps compare several polar models during one study session or class demo.

FAQs

What does this calculator convert?

It converts common polar points and polar equations into rectangular coordinate form. It supports circles, lines, rose forms, conic style equations, limacons, and custom substitution guidance.

Which formulas are used most often?

The main identities are x = r cos theta, y = r sin theta, and r squared equals x squared plus y squared. Other identities are derived from these.

Can it convert polar points?

Yes. Select point mode, enter r and theta, choose the angle unit, and submit. The calculator returns the rectangular point using cosine and sine.

Why do some answers include radicals?

Some polar equations contain r after substitution. Since r equals the square root of x squared plus y squared, a radical may remain before final simplification.

Does squaring change the equation?

Squaring can introduce extra points. When a radical equation is squared, compare the final equation with the original polar rule to confirm restrictions.

Should I use degrees or radians?

Use degrees for angle values like 30, 45, or 180. Use radians for decimal radian values such as 0.7854 or 3.1416.

Can I save the result?

Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a simple printable result summary.

Does custom mode fully simplify every equation?

No. Custom mode gives the correct substitution identities and guidance. Very complex equations may still need manual algebra after the identities are applied.