Calculator Input
Example Data Table
| x | y | Radius r | Angle θ | Polar Form |
|---|---|---|---|---|
| 3 | 4 | 5 | 53.1301° | 5∠53.1301° |
| -5 | 12 | 13 | 112.6199° | 13∠112.6199° |
| 2 | -2 | 2.8284 | 315° | 2.8284∠315° |
Formula Used
The calculator converts a rectangular point or complex value into polar form.
Radius: r = √(x² + y²)
Angle: θ = atan2(y, x)
Polar pair: (r, θ)
Complex polar form: z = r(cos θ + i sin θ)
Exponential form: z = reiθ
How To Use This Calculator
- Enter the x value or real part.
- Enter the y value or imaginary part.
- Select degrees or radians for the main answer.
- Choose a positive or signed angle range.
- Set decimal precision.
- Add optional batch pairs in x,y format.
- Press the convert button.
- Download the answer as CSV or PDF.
Converting Rectangular Values To Polar Form
What polar form means
Polar form describes a point by distance and direction. Rectangular form uses horizontal and vertical movement. Polar form uses radius and angle instead. This makes rotation, complex multiplication, and wave calculations easier. A point like (3, 4) becomes a radius of 5 with an angle near 53.13 degrees.
Why this conversion matters
Many fields use polar form. It appears in electronics, signals, robotics, physics, navigation, and complex numbers. Rectangular form is simple for grid position. Polar form is better when direction and magnitude matter. It also helps when values rotate around an origin.
Understanding the radius
The radius is the straight distance from the origin. It is always non-negative. The calculator finds it with the Pythagorean theorem. It squares x and y, adds them, and takes the square root. This gives the length of the vector.
Understanding the angle
The angle shows the direction from the positive x-axis. A normal tangent formula can miss the correct quadrant. This calculator uses atan2. It reads both x and y. That gives a safer angle for every quadrant and axis case.
Degrees and radians
Degrees are common in classroom work. Radians are common in advanced mathematics and programming. This tool shows both. You can select the main display unit. You can also choose a signed angle or a positive angle range.
Complex number use
A complex number a + bi can be treated like a point. The real part is x. The imaginary part is y. Its polar form is useful for powers, roots, multiplication, and division. Exponential notation is also shown for advanced work.
Batch conversion
The batch box accepts several pairs. Place one x,y pair on each line. The calculator creates a table for all valid rows. This is useful for homework, engineering notes, and reports. The same scale and precision settings apply to all rows.
Graph reading
The graph draws the vector from the origin to the converted point. It helps confirm direction visually. The plotted marker shows the final point. The line shows the radius direction. This makes quadrant errors easier to notice.
FAQs
1. What is polar form?
Polar form describes a point or complex number using radius and angle. The radius gives distance from the origin. The angle gives direction from the positive x-axis.
2. What does r mean?
r means radius or magnitude. It is the distance from the origin to the point. It is calculated with √(x² + y²).
3. What does theta mean?
Theta is the angle of the point or vector. It measures direction from the positive x-axis. It can be shown in degrees or radians.
4. Why use atan2 instead of tangent?
atan2 uses both x and y values. It identifies the correct quadrant. A simple tangent inverse may give an angle in the wrong quadrant.
5. Can this convert complex numbers?
Yes. Enter the real part as x. Enter the imaginary part as y. The result shows polar, trigonometric, and exponential forms.
6. What happens at the origin?
At the origin, the radius is zero. The angle is usually treated as undefined. The calculator labels the point as the origin.
7. Which angle range should I choose?
Use positive range for angles from 0 to 360 degrees. Use signed range when you want angles from -180 to 180 degrees.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report with the main polar result.