Polar to Cartesian Coordinates
Polar coordinates say “how far, which way”. x is radius × cos(angle) and y is radius × sin(angle). It is how you plot points on a circle or position a robot arm.
Your numbers
x
8.1915
y
5.7358
Radians
0.6109
Formula
x = Radius r × cos Angle θ, y = Radius r × sin Angle θ
The angle is measured from the positive x-axis, counter-clockwise positive. Compass bearings use a different origin and direction.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Radius rThe distance from the origin to the point. | 10 | 0 and up |
| Angle θ (°)The angle from the positive x-axis, counter-clockwise. | 35 | -360 ~ 360 |
Step by step
Quick reference table
Results when only Radius r changes and everything else stays put.
| Radius r | x | y | Radians |
|---|---|---|---|
| 5 | 4.0958 | 2.8679 | 0.6109 |
| 7.5 | 6.1436 | 4.3018 | 0.6109 |
| 10 | 8.1915 | 5.7358 | 0.6109 |
| 15 | 12.2873 | 8.6036 | 0.6109 |
| 20 | 16.383 | 11.4715 | 0.6109 |
What each result means
| Result | At default values |
|---|---|
| xThe horizontal position in Cartesian form. | 8.1915 |
| yThe vertical position in Cartesian form. | 5.7358 |
| RadiansAn angle in radians; a full turn is 2π. | 0.6109 |
Common mistakes
The angle is measured from the positive x-axis, counter-clockwise positive. Compass bearings use a different origin and direction.
Glossary
- Radius r
- The distance from the origin to the point.
- Angle θ
- The angle from the positive x-axis, counter-clockwise.
- x
- The horizontal position in Cartesian form.
- y
- The vertical position in Cartesian form.
- Radians
- An angle in radians; a full turn is 2π.
Frequently asked questions
Q. How is Polar to Cartesian Coordinates calculated?
x = Radius r × cos Angle θ, y = Radius r × sin Angle θ — Polar coordinates say “how far, which way”. x is radius × cos(angle) and y is radius × sin(angle). It is how you plot points on a circle or position a robot arm.
Q. Can you walk through an example?
With Radius r 10, Angle θ 35°, the answer is x 8.1915.
Q. What do I need to enter?
Enter Radius r, Angle θ. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Radius r moves the answer to Radius r 5 → x 4.0958 and Radius r 20 → x 16.383. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
The angle is measured from the positive x-axis, counter-clockwise positive. Compass bearings use a different origin and direction.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.