Polar to Cartesian Equation Calculator

Calculator Input

Conversion type

Radius or constant r

Theta for point

Angle unit

Fixed theta for line

Coefficient a

Coefficient b

Integer n for rose curve

Rose function

Decimal precision

Custom polar expression

Formula Used

The calculator uses these core identities:

x = r cos(theta)
y = r sin(theta)
r² = x² + y²
tan(theta) = y / x
cos(theta) = x / r
sin(theta) = y / r

For a polar point, the first two formulas give the coordinate pair. For equations, the tool replaces polar parts with Cartesian identities and simplifies standard curve forms.

How to Use This Calculator

Select the conversion type that matches your polar point or equation.
Enter r, theta, a, b, n, or the custom expression as needed.
Choose degrees or radians for angle based entries.
Set decimal precision for point results and coefficients.
Press Submit to view the result above the form.
Use the CSV or PDF button to save the current result.

Example Data Table

Polar input	Type	Cartesian output	Note
r = 5, theta = 45°	Point	(3.5355, 3.5355)	Uses x = r cos(theta), y = r sin(theta)
r = 6	Circle	x² + y² = 36	Circle centered at origin
theta = 30°	Line	y = 0.5774x	Line through origin
r = 4 cos(theta)	Shifted circle	x² + y² = 4x	Center is (2, 0)
r = 3 + 2 sin(theta)	Limaçon	(x² + y² - 2y)² = 9(x² + y²)	Squared Cartesian form

Understanding Polar Conversion

Polar equations describe locations by distance and angle. Cartesian equations describe them with horizontal and vertical coordinates. This calculator connects both views. It is useful for algebra, calculus, graphing, engineering, navigation, and curve study. A point uses r and theta. An equation uses relationships between r, sine, cosine, and theta.

Why Cartesian Form Helps

Cartesian form is often easier to graph on standard axes. It also helps compare curves with circles, lines, and conic shapes. For example, r equals a constant becomes a circle centered at the origin. The equation r equals a times cosine theta becomes a circle shifted along the x axis. The sine version shifts along the y axis. These patterns are easier to see after conversion.

How The Calculator Works

The tool applies core identities. It uses x equals r cosine theta. It uses y equals r sine theta. It also uses r squared equals x squared plus y squared. When a polar equation contains cosine theta, the tool can replace it with x over r. When it contains sine theta, it can replace it with y over r. Then it simplifies the expression when a standard form is available.

Common Equation Types

Some polar equations convert to clean algebraic equations. Others keep an angle term. Rose curves with larger integer values may need atan2 in Cartesian form. Spirals also use atan2 because the angle remains part of the rule. The calculator explains this when a pure polynomial equation is not practical.

Accuracy And Limits

Exact algebra works best for standard families. Decimal point results depend on selected precision. Very small rounding differences can appear after trigonometric steps. Treat the answer as a guide, not as a replacement for teacher instructions. When your class requires a form, rewrite the final equation to match that form.

Practical Study Tips

Always check the angle unit before converting a point. Degrees must be changed into radians for trigonometric functions. Negative radius values are allowed in polar work, but they can surprise beginners. They point in the opposite direction from the given angle. For equation work, review every squared step. Squaring can sometimes add extra points. Use the final note and sample table as a quick check.

FAQs

What does polar to Cartesian mean?

It means changing a point or equation from r and theta notation into x and y notation. The new form can be graphed on standard rectangular axes.

Which formulas are used most often?

The most common formulas are x = r cos(theta), y = r sin(theta), and r² = x² + y². Equations also use sine, cosine, and tangent substitutions.

Can this calculator convert points?

Yes. Choose the point option, enter r and theta, select the angle unit, and submit. The result gives the matching x and y coordinates.

Can every polar equation become a simple Cartesian equation?

No. Some curves keep an angle term after conversion. Rose curves and spirals can require atan2 notation or advanced algebraic expansion.

Why does a limaçon result use a squared form?

A limaçon has a term like a plus b cosine theta. After multiplying by r, one side still contains r. Squaring removes the radical.

Does angle unit matter?

Yes. Point conversion depends on trigonometric values. Enter degrees as degrees and radians as radians. A wrong unit gives a different coordinate pair.

What happens when theta creates a vertical line?

When theta has no valid tangent slope, the calculator returns x = 0. This represents a vertical line through the origin.

Can I save the result?

Yes. After submitting the form, use the CSV button for spreadsheet data or the PDF button for a simple report copy.