Polar to Rectangular Coordinates Calculator
A point in polar coordinates is described by its distance from the origin, r, and the angle θ it makes with the positive x-axis. Converting to rectangular (Cartesian) coordinates uses x = r·cos θ and y = r·sin θ. Enter the radius and angle (in degrees) to get x and y.
Why x = r·cos θ and y = r·sin θ
Drop a perpendicular from the point to the x-axis and you get a right triangle with hypotenuse r and angle θ at the origin. The adjacent side is x = r·cos θ and the opposite side is y = r·sin θ — the same right-triangle trig ratios used everywhere else, just applied to a point instead of a triangle's side.
Worked Example
Take r=5, θ=60°. x = 5×cos(60°) = 5×0.5 = 2.5, and y = 5×sin(60°) = 5×0.8660 ≈ 4.33. The point (2.5, 4.33) sits 5 units from the origin at a 60° angle from the positive x-axis — you can check the distance with √(2.5²+4.33²) ≈ 5, matching r.
Quadrant Signs Follow the Angle
Between 0° and 90°, both x and y come out positive (first quadrant). Between 90° and 180°, cos θ turns negative while sin θ stays positive, so x is negative and y positive (second quadrant) — the signs of cos and sin at each angle tell you which quadrant the rectangular point lands in before you even compute the numbers.
Frequently Asked Questions
What's the formula to convert polar to rectangular coordinates?
x = r·cos θ and y = r·sin θ, where r is the distance from the origin and θ is measured counterclockwise from the positive x-axis. Both formulas come directly from the right-triangle definitions of sine and cosine.
How do I convert rectangular back to polar coordinates?
r = √(x²+y²) and θ = atan2(y, x), using the two-argument arctangent to get the correct angle in the right quadrant — a plain arctan(y/x) alone can't distinguish, for example, (1,1) from (−1,−1).
What does a negative angle mean?
A negative θ measures clockwise from the positive x-axis instead of counterclockwise. −90° and 270° land on the same point, since angles repeat every 360°.
What point does r=0 give?
The origin, (0,0), no matter what angle θ you enter — when the radius is zero, the angle becomes meaningless since every angle points to the same spot.
Does this work with angles in radians?
This calculator expects degrees. To convert radians to degrees first, multiply by 180/π — for example, π/4 radians is 45°.