What this polar to rectangular calculator does
This convert polar to rectangular calculator transforms a point from polar coordinates into rectangular coordinates (also called Cartesian coordinates). If your point is given as (r, θ), the calculator returns the equivalent (x, y).
It is useful for trigonometry, precalculus, engineering graphics, physics vectors, and complex number work. You can choose degrees or radians, set decimal precision, and get immediate results.
Polar to rectangular formula
To convert polar coordinates to rectangular coordinates, use:
x = r cos(θ)
y = r sin(θ)
Where:
- r = distance from the origin (radius)
- θ = angle from the positive x-axis
- x, y = rectangular coordinate values
How to use the calculator
- Enter the radius r.
- Enter the angle θ.
- Pick the angle unit: degrees or radians.
- Select decimal places.
- Click Convert.
Tip: You can enter negative angles and negative radius values. The calculator handles both correctly.
Worked examples
Example 1: (5, 53.13°)
Using x = 5 cos(53.13°) and y = 5 sin(53.13°), you get approximately:
- x ≈ 3
- y ≈ 4
So the rectangular coordinate is (3, 4).
Example 2: (10, 210°)
Since 210° is in Quadrant III, both x and y are negative.
- x = 10 cos(210°) ≈ -8.6603
- y = 10 sin(210°) = -5
Result: (-8.6603, -5)
Example 3: (3, 1.2 rad)
In radians mode:
- x = 3 cos(1.2) ≈ 1.0871
- y = 3 sin(1.2) ≈ 2.7961
| Polar Input | Unit | Rectangular Output |
|---|---|---|
| (5, 53.13) | Degrees | (3.0000, 4.0000) |
| (10, 210) | Degrees | (-8.6603, -5.0000) |
| (3, 1.2) | Radians | (1.0871, 2.7961) |
Why convert polar coordinates to rectangular?
Polar form is great when magnitude and direction are natural inputs (like vectors or phasors). Rectangular form is better for plotting points, graphing equations, and doing many algebraic operations.
- Engineering: Convert phasor form to a + bi form.
- Physics: Break force vectors into horizontal and vertical components.
- Math classes: Move between coordinate systems for graphing and analysis.
- Signal processing: Convert magnitude-angle data into in-phase and quadrature components.
Common mistakes to avoid
- Wrong angle unit: Using degrees in radian mode (or vice versa).
- Swapping sine and cosine: x uses cosine, y uses sine.
- Rounding too early: Keep more digits in intermediate steps when needed.
- Quadrant sign errors: Check if x and y should be positive or negative based on θ.
Quick FAQ
Can radius be negative?
Yes. A negative radius points in the opposite direction of the angle. This calculator supports it.
Does this work for complex numbers?
Yes. Polar complex numbers r(cosθ + i sinθ) convert to x + yi using the same formulas.
What if the result shows -0.0000?
Very tiny floating-point values are normalized to 0 by the calculator to keep output clean.
Final thoughts
If you need a fast and accurate way to convert polar form to rectangular form, this page gives you both the calculator and the method. Use it to solve homework, verify manual calculations, or speed up engineering work. Bookmark it for quick polar to Cartesian conversion anytime.