Coordinate converter
Convert Polar Coordinates
Use a standard counterclockwise angle or reverse it for clockwise measurements. Negative radii are supported.
Example Data Table
These examples use standard counterclockwise angles and four decimal places.
| Radius r | Angle | Unit | x = r cos θ | y = r sin θ | Location |
|---|---|---|---|---|---|
| 1 | 45 | Degrees | 0.7071 | 0.7071 | Quadrant I |
| 5 | 120 | Degrees | -2.5000 | 4.3301 | Quadrant II |
| 3 | 0.5236 | Radians | 2.5981 | 1.5000 | Quadrant I |
| -4 | 0 | Degrees | -4.0000 | 0.0000 | Negative x-axis |
Formula Used
A polar point has radius r and angle θ. Convert the angle to radians before applying trigonometric functions.
θradians = θ × π / 180 for degreesθradians = θ × π / 200 for gradiansx = r cos(θ)y = r sin(θ)
When Clockwise is selected, the calculator changes the angle sign first. Standard mathematical angles run counterclockwise from the positive x-axis.
How to Use This Calculator
- Enter the radius r. Use a negative radius only when your problem provides one.
- Enter the angle value exactly as given.
- Select degrees, radians, or gradians.
- Choose counterclockwise for standard trigonometry. Choose clockwise for reversed measurements.
- Set the number of decimal places you want to display.
- Select Calculate Coordinates. Read the ordered pair above the form.
- Copy the result, download the CSV file, or save the result as a PDF.
Polar Coordinates Made Practical
Polar coordinates describe a point by distance and direction. The distance is called the radius. The direction is called the angle. Together, they identify one point from the origin. This format is useful when rotation, circles, waves, or repeated directions matter.
Rectangular coordinates use horizontal and vertical distances. They write the same point as an ordered pair, x and y. The x value measures movement across the graph. The y value measures movement up or down. Converting between both systems makes many coordinate problems easier to check.
Why Convert Polar Values
Polar form is compact for circular motion. It also suits vectors and complex numbers. However, graphing tools and algebra problems often need x and y values. Rectangular form helps you plot a point on standard axes. It also supports distance, slope, and equation calculations.
This calculator turns a radius and angle into rectangular coordinates. It accepts degrees, radians, and gradians. It also normalizes the entered angle. Normalization makes the direction easier to compare. A negative radius remains valid. It represents the opposite direction from the angle entered.
Reading the Result
The result shows the rectangular pair. It also displays the converted angle in radians. Radians are required by the cosine and sine functions used internally. The calculator then shows the normalized direction. This is the matching angle within one complete turn.
Quadrant information is included for quick checking. A point can lie in Quadrant I, II, III, or IV. Some points fall directly on an axis. The origin is another special case. These details help you detect sign errors before using the values elsewhere.
Accuracy and Rounding
Real coordinate values can include long decimals. Select the precision you need before calculating. Lower precision keeps results readable. Higher precision helps with engineering, trigonometry, and later calculations. Rounding only changes the displayed result. The calculator keeps the computed value for its current result.
Use enough digits when another calculation follows. Early rounding can create a visible difference. This is important near axes. A small y value can appear as zero after rounding. Check the full precision when the direction looks unexpected.
Common Uses
Students use polar conversion for trigonometry exercises. Engineers use it for forces and rotating parts. Programmers use it for movement around a center point. Designers use it for radial layouts. Scientists use it for directional measurements. The same core conversion works in every case.
Start with the unit shown beside the angle. Enter the radius exactly as given. Select an appropriate precision. Then compare the signs of x and y with the expected quadrant. A positive angle moves counterclockwise from the positive x-axis. A negative angle moves clockwise.
Keep the original polar form with the rectangular answer. Both forms describe the same location. This makes later checking simple. It also reduces confusion when an exercise switches coordinate systems for every learner and professional user.
Frequently Asked Questions
What does polar to rectangular conversion do?
It changes a point written as radius and angle into horizontal and vertical coordinates. The position remains identical. Only the coordinate system changes.
Can I enter angles in degrees?
Yes. Select Degrees before calculating. The calculator converts the value to radians internally, then applies cosine and sine.
Why are radians shown in the result?
Trigonometric functions use radians in their standard computational form. Showing the converted value makes the calculation easier to audit and reuse.
What does a negative radius mean?
A negative radius is valid. It places the point in the direction opposite the entered angle. The displayed x and y values already reflect that reversal.
How does the clockwise setting work?
Standard polar angles turn counterclockwise from the positive x-axis. Choose Clockwise when your angle is measured in the opposite direction. The calculator adjusts its sign before conversion.
What is a normalized angle?
It is an equivalent angle placed within one complete turn. The range is 0 to 360 degrees, 0 to 2π radians, or 0 to 400 gradians.
Why does an output show zero?
The coordinate may be exactly zero, or it may round to zero at the selected precision. Increase decimal places to inspect a small value.
Can this calculator help with complex numbers?
Yes. A polar complex number r(cos θ + i sin θ) converts to x + yi. The displayed x is the real part, and y is the imaginary coefficient.
Which precision should I choose?
Use two to four decimals for routine homework and graphing. Choose more decimals for engineering, scientific work, or results used in later calculations.
Does the calculator save my values?
The current result remains visible after calculation. Copy the summary or download the CSV file for a record. Refreshing or leaving the page clears the form.
How can I verify the coordinates?
Check signs against the expected quadrant. Then substitute x and y into x² + y² = r². Calculate coordinates confidently, then use them in your work.