Calculate rectangular coordinates
Enter one polar coordinate pair. Negative radii are supported.
Formula used
Use these equations to change a polar point into a rectangular point.
Here, r is the radius and θ is the angle. The final ordered pair is written as (x, y).
How to use this calculator
- Enter the radius from your polar coordinate.
- Enter the angle value exactly as given.
- Select degrees, radians, or gradians.
- Choose the decimal precision you need.
- Press convert and review the pair, steps, and location.
Example polar coordinate conversions
These examples show common angle positions. Values are rounded to four decimal places.
| Radius | Angle | Unit | x coordinate | y coordinate | Location |
|---|---|---|---|---|---|
| 10 | 30 | Degrees | 8.6603 | 5.0000 | Quadrant I |
| 8 | 90 | Degrees | 0.0000 | 8.0000 | Positive y-axis |
| 12 | 210 | Degrees | -10.3923 | -6.0000 | Quadrant III |
| 5 | 1.5708 | Radians | 0.0000 | 5.0000 | Positive y-axis |
Understanding polar and rectangular coordinates
Two useful coordinate systems
Polar coordinates describe a location with distance and direction. The distance is called the radius. The direction is called the angle. Rectangular coordinates describe the same location with horizontal and vertical distances. They use an ordered pair written as x and y. Converting between these systems is common in geometry, graphics, surveying, navigation, physics, and engineering. A reliable conversion helps you plot points correctly. It also helps you compare values across diagrams and formulas.
Read the radius and angle carefully
The radius tells how far the point lies from the origin. A positive radius moves outward along the selected angle. A negative radius represents the opposite direction. The angle sets the point's rotation from the positive horizontal axis. Degrees are familiar in school geometry. Radians are common in calculus and programming. Gradians are sometimes used in surveying. Choose the same unit that appears in the original polar coordinate. A wrong angle unit can produce a completely different rectangular point.
Use sine and cosine
The conversion uses trigonometry. Cosine finds the horizontal component. Sine finds the vertical component. Multiply each component by the radius. The results become the x and y values. For example, a radius of ten at thirty degrees creates a positive horizontal value and a positive vertical value. The point lies in the first quadrant. At one hundred fifty degrees, x becomes negative while y remains positive. The point then lies in the second quadrant.
Choose practical precision
Precision matters when the angle does not make a simple special triangle. Many calculations create long decimal values. Select a practical number of decimal places before saving results. Four places work well for general homework and technical notes. More places can help when later calculations are sensitive. Fewer places make reports easier to read. Round only after the calculator completes the conversion. Early rounding may change later measurements or plotted positions.
Check the displayed point
The result card shows the rectangular pair and the converted angle in radians. It also identifies the point's position. This makes checking easier. Points on an axis have one coordinate equal to zero. Points in a quadrant have positive or negative coordinate signs. Compare those signs with the original angle. This quick review catches many entry errors. A radius of zero always converts to the origin. In that case, the angle does not affect the final rectangular values.
Use results in later work
Use the calculator when solving coordinate geometry problems. It is also useful for vector components, circular motion, complex numbers, robotic arms, and map bearings. Keep units consistent across your project. Copy the resulting pair into a graph or equation when needed. Export the result as a CSV file for records. You can also print the page and save it as a PDF. Review the formula steps before submitting important work. Clear inputs and repeat the process for another point. This approach supports quick checks without losing mathematical meaning. Always label coordinates clearly. Good labels prevent confusion when several converted points appear on one drawing.
Frequently asked questions
1. What does polar to rectangular conversion mean?
It changes a point written as radius and angle into horizontal and vertical coordinates. The same physical location is represented as the ordered pair (x, y).
2. Which formulas does the calculator use?
It uses x = r × cos(θ) and y = r × sin(θ). The angle is converted to radians first when you enter degrees or gradians.
3. Should I choose degrees, radians, or gradians?
Choose the unit used by your original angle. Use degrees for most geometry questions. Use radians for calculus and many programming tasks. Use gradians when the source provides gon values.
4. Can the radius be negative?
Yes. A negative radius is valid in polar coordinates. It places the point in the direction opposite the entered angle. The calculator applies this rule automatically.
5. What happens when the radius is zero?
The point becomes the origin, written as (0, 0). The chosen angle does not change that result because there is no distance from the origin.
6. Why can small decimal values appear as zero?
Values close to zero may result from floating-point arithmetic. The selected precision rounds them for clean display. Increase decimal places when you need to inspect a very small value.
7. Why do coordinate signs change by quadrant?
Cosine controls the x sign, while sine controls the y sign. Their signs change as the angle moves around the coordinate plane. This determines the quadrant.
8. Can I use this for vector components?
Yes. Treat the vector magnitude as the radius and its direction as the angle. The x and y outputs become the horizontal and vertical vector components.
9. What is an ordered pair?
An ordered pair is written as (x, y). The first value gives horizontal position. The second value gives vertical position. Order matters because (x, y) differs from (y, x).
10. What is included in the CSV download?
The CSV file includes your radius, input angle, unit, radians value, x coordinate, y coordinate, and the identified axis or quadrant location.
11. How do I save the calculation as a PDF?
Press Print / Save PDF after calculating. Your browser print dialog opens. Choose a PDF destination, then save the result with its calculation steps and reference sections.