Calculator Form
Formula Used
A polar coordinate uses a radius and an angle. The radius is the distance from the origin. The angle shows the direction from the positive x-axis. To convert polar form into rectangular form, use these formulas.
x = r cos(θ)
y = r sin(θ)
Here, r is the radius. θ is the angle. The angle must be handled in radians for most programming functions. This calculator converts degrees and gradians before applying trigonometry.
How To Use This Calculator
Enter the radius in the first field. Then enter the angle value. Choose whether the angle is in degrees, radians, or gradians. Select the number of decimal places needed. Use the scale field when coordinates must be enlarged or reduced. Use offset fields when a shifted origin is required. Press the calculate button. The result appears below the header and above the form.
Example Data Table
| Radius | Angle | Unit | x | y | Coordinate |
|---|---|---|---|---|---|
| 10 | 0 | Degrees | 10 | 0 | (10, 0) |
| 10 | 90 | Degrees | 0 | 10 | (0, 10) |
| 8 | 45 | Degrees | 5.6569 | 5.6569 | (5.6569, 5.6569) |
| 6 | 3.1416 | Radians | -6 | 0 | (-6, 0) |
Polar To Rectangular Coordinate Guide
What The Calculator Does
This polar to rectangular coordinates calculator changes a point from distance-angle form into x-y form. Polar form is written as r and θ. Rectangular form is written as x and y. Both forms describe the same point. They simply use different references. Polar form starts with a distance from the origin. It then uses an angle to set direction. Rectangular form uses horizontal and vertical distances.
Why Conversion Matters
This conversion is useful in algebra, trigonometry, physics, surveying, navigation, robotics, mapping, and signal work. Many real problems begin with a distance and direction. Graphing software and spreadsheets often need x and y values. This tool bridges those formats. It also shows the angle in degrees and radians. That makes checking work easier.
Advanced Options
The calculator supports degrees, radians, and gradians. It also includes decimal precision control. A scale factor can enlarge or reduce the coordinate. Offset fields can shift the final point from a new local origin. These options help when points belong to drawings, models, maps, layouts, or transformed coordinate systems.
Interpreting The Result
The x value shows horizontal movement. Positive x moves right. Negative x moves left. The y value shows vertical movement. Positive y moves upward. Negative y moves downward. The quadrant label explains the point location. The slope gives the ratio of vertical change to horizontal change. If x is zero, slope is undefined.
Accuracy Notes
Trigonometric results may contain very small rounding differences. That is normal in decimal calculations. Increase decimal places when precision matters. Use fewer places when preparing simple classroom answers. Always match the selected angle unit to your source data. A wrong unit gives a wrong coordinate.
FAQs
What is a polar coordinate?
A polar coordinate describes a point using radius and angle. The radius gives distance from the origin. The angle gives direction from the positive x-axis.
What is a rectangular coordinate?
A rectangular coordinate describes a point using x and y values. The x value is horizontal position. The y value is vertical position.
How do I convert polar to rectangular?
Use x = r cos θ and y = r sin θ. Make sure the angle unit is correct before calculating the trigonometric values.
Can the radius be negative?
Yes. A negative radius places the point in the opposite direction from the entered angle. The calculator still applies the same formulas.
Why does the calculator show radians?
Most programming trigonometry functions use radians. Showing radians helps verify the internal angle conversion and supports advanced math work.
What does scale factor mean?
The scale factor multiplies the calculated x and y values. It is helpful for drawings, maps, models, and coordinate transformations.
What are offset x and offset y?
Offsets shift the final coordinate. They are useful when the origin is not at zero or when working with local coordinate systems.
Why is slope sometimes undefined?
Slope is y divided by x. When x is zero, division by zero is not valid. Therefore, the slope is undefined.