Polar and Rectangular Calculator

Calculator Form

Choose a direction. Fill rectangular values for x and y, or polar values for radius and angle.

Conversion direction

Rectangular x

Rectangular y

Polar radius r

Polar angle theta

Polar angle unit

Output angle unit

Scale multiplier

Rotation angle

Rotation unit

Decimal precision

Angle handling

Normalize final angle

Reset

Example Data Table

Input Type	Input Values	Expected Main Result	Use Case
Rectangular	x = 3, y = 4	r = 5, theta = 53.1301°	Basic vector check
Polar	r = 10, theta = 30°	x = 8.6603, y = 5	Direction to components
Rectangular	x = -5, y = 12	r = 13, theta = 112.6199°	Quadrant verification
Polar	r = 7, theta = 0.785398 rad	x = 4.9497, y = 4.9497	Radian conversion

Formula Used

Rectangular to polar:

r = √(x² + y²)

theta = atan2(y, x)

Polar to rectangular:

x = r cos(theta)

y = r sin(theta)

Scaling:

x scaled = x × scale

y scaled = y × scale

Rotation:

x final = x scaled cos(phi) - y scaled sin(phi)

y final = x scaled sin(phi) + y scaled cos(phi)

The calculator uses atan2 because it preserves quadrant direction better than a single tangent inverse expression.

How to Use This Calculator

Select rectangular to polar or polar to rectangular.
Enter x and y values for rectangular conversion.
Enter radius and angle for polar conversion.
Choose degrees or radians for angle input.
Select the angle unit wanted in the answer.
Add an optional scale multiplier.
Add an optional rotation angle.
Choose rounding precision and press calculate.
Use CSV or PDF buttons to save the result.

Understanding Polar and Rectangular Coordinates

Polar and rectangular coordinates describe the same point in different ways. Rectangular form uses horizontal x and vertical y values. Polar form uses a radius r and an angle theta. This calculator helps you move between both forms without losing important detail.

Rectangular coordinates are common in graphs, maps, drawings, and spreadsheets. They show how far a point moves left or right, then up or down. Polar coordinates are useful when direction matters. They appear in navigation, electronics, surveying, and vector work.

Why This Conversion Matters

A clean conversion prevents sign errors and quadrant mistakes. The angle depends on both x and y, so a simple inverse tangent is not enough. The atan2 method checks the correct quadrant and returns a dependable direction. The radius then shows distance from the origin.

When you enter polar values, the calculator uses cosine for the x component. It uses sine for the y component. Degree and radian choices make the tool flexible. Rounding controls also help you match class notes, reports, or technical sheets.

Advanced Options

The scale field changes vector length while keeping direction before rotation. The rotation field turns the vector by a chosen angle. These options make the calculator useful for transformed coordinates, rotated axes, and quick vector checks. The final output includes rectangular, polar, complex, and bearing style results.

The normalized angle option keeps degrees between zero and three hundred sixty. In radians, it keeps the angle between zero and two pi. This makes reports easier to compare. Unnormalized angles can still be useful when direction changes over repeated turns.

Practical Uses

Students can verify homework steps. Engineers can check vectors before drawing. Surveyors can compare distance and direction. Programmers can prepare coordinate logic for games, charts, or simulations. The export buttons save a simple record for later review.

Always check your unit choice first. A degree entered as radians creates a very different point. Also confirm the sign of x and y values. Negative components can move the answer into another quadrant. With careful inputs, this tool gives clear, repeatable, and easy to audit coordinate results.

Use the example table to compare typical inputs before entering your own values and chosen precision settings.

FAQs

What does this calculator convert?

It converts rectangular coordinates into polar coordinates and polar coordinates into rectangular coordinates. It also supports scaling, rotation, angle normalization, complex form, and export options.

What is rectangular form?

Rectangular form uses x and y values. The x value measures horizontal position. The y value measures vertical position. Together, they define a point or vector.

What is polar form?

Polar form uses radius and angle. The radius gives distance from the origin. The angle gives direction from the positive x-axis.

Why does the calculator use atan2?

atan2 checks both x and y signs. This helps place the angle in the correct quadrant. It avoids common direction mistakes.

Can I use radians?

Yes. You can enter polar angles in degrees or radians. You can also choose degrees or radians for the final angle output.

What does normalize final angle mean?

It adjusts the final angle into a standard positive range. Degrees become zero to three hundred sixty. Radians become zero to two pi.

How does rotation work?

Rotation turns the scaled vector around the origin. Positive angles rotate counterclockwise in the standard coordinate plane.

Can I save my result?

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