Polar to cartesian coordinates
The relation
x
A point at distance r and angle v from the positive x-axis lies at x = r cos v and y = r sin v.
Solve for
x=5
x equals 10 times the cosine of (60 degrees), equals 10 times 0.5, equals 5
Definition — what does that mean?
Polar to cartesian coordinates
A point at distance r and angle v from the positive x-axis lies at x = r cos v and y = r sin v.
In Swedish: Polära till kartesiska koordinater
Practice questions
You are given
- r=5
- v=180
Round to 2 decimals.
The formula
- x = r cos v
- y = r sin v
r is the distance from the origin and v the angle measured counter-clockwise from the positive x-axis. The card above computes x; y is the same with sine, and the coordinate converter gives both.
A worked example
r = 10 and v = 60°: x = 10 cos 60° = 10 · 0.5 = 5 and y = 10 sin 60° = 10 · 0.866 = 8.66. At v = 150° the same r gives x = 10 · (−0.866) = −8.66 and y = 5: the point has moved into the second quadrant and x has changed sign.
The unit circle behind it
On the unit circle, the point at angle v is (cos v, sin v) by definition. A point at distance r is that point scaled by r, which is the whole formula. The signs of cos and sin in each quadrant come with it: cos is negative left of the y-axis, sin below the x-axis.
Degrees or radians
The formula is the same; the calculator has to be in the mode the angle is given in. 10 cos 60 in radian mode is −9.52, not 5, and the mistake looks like a plausible coordinate.
Common mistakes
- Swapping sine and cosine. x goes with cosine, the horizontal shadow.
- Measuring the angle clockwise or from the y-axis. Convention is counter-clockwise from the positive x-axis; a bearing from north needs 90° − bearing first.
- A negative r. The distance is taken positive; a negative r is the same point at v + 180°.
Frequently asked questions
How do I go the other way?
r = √(x² + y²) and the angle from atan2(y, x), on the
cartesian to polar page.
Is this the same as a complex number’s polar form?
Yes: r(cos v + i sin v) is the complex number x + iy with the same r and v, on the
complex number forms page.
What about three dimensions?
Cylindrical coordinates add a z; spherical ones add a second angle. The plane formula is the first
step of both.
Looking for another letter? The formula solved for each one
| Solve for | Rearranged | With the numbers | Comes to |
|---|---|---|---|
| r | xcos(v) | 5cos(60°) | 10 |
| x | r cos(v) | 10 cos(60°) | 5 |
Last reviewed October 20, 2026 ·How pages are reviewed
Calculate it in practice
Related formulas
Builds on
Put this formula on your site
Copy the code into your page where the formula card should appear. It loads from lovetocalc.com, so it stays current on its own — all we ask is that the credit line comes with it.