When should I use polar coordinates instead of rectangular?

Polar coordinates are ideal when dealing with circular or rotational problems, such as describing orbits, spirals, or any situation where distance from a center point and angle are more natural measurements than x and y positions.

Can the radius be negative?

Yes, a negative radius means the point is in the opposite direction. The point (-r, θ) is the same as (r, θ + 180°). This convention is sometimes used in mathematics but can be confusing, so positive radii are more common.

How do I convert degrees to radians?

Multiply degrees by π/180. For example, 90° = 90 × π/180 = π/2 radians. Conversely, multiply radians by 180/π to get degrees.

What's the relationship between polar form and complex numbers?

A complex number x + yi corresponds to the point (x, y) in rectangular coordinates, which equals (r, θ) in polar form where r = √(x² + y²) and θ = atan2(y, x). This is written as r∠θ or r·e^(iθ).

Polar to Rectangular Calculator – Convert Coordinates

Convert polar coordinates to rectangular (Cartesian) form

Method and source check

Last checked: Sep 30, 2026

Automated source and implementation review, including worked examples and documented numerical limits. This review does not provide professional advice or specialist approval.

Method

Convert the selected angle unit to radians, then multiply radius by cosine and sine to obtain Cartesian x and y. Negative radii are allowed by the coordinate convention.

x = r cosθ

y = r sinθ

θ_rad = θ_deg × π/180

Variables and units

r: signed radial coordinate
θ: angle measured from positive x-axis
x,y: Cartesian coordinates

Worked example

For r=2 and θ=90°, x=0 and y=2; the same result follows from θ=π/2 radians.

Limitations

Displayed x,y and radian angle are rounded to six decimals; degree display is rounded to two.

Rounding can make near-zero trigonometric components display as zero.
Reverse conversion uses atan2(y,x) and is explanatory only; this calculator converts polar to Cartesian.
Unsupported derived numerical ranges are rejected with a localized error on a visible form field; the calculations use floating-point arithmetic.
(−2,0°) and (2,180°) represent the same point.
At r=0 all angles represent the origin.

Sources

Polar Coordinates, Calculus Volume 3 — OpenStax(opens in a new tab)

Table of Contents

How to Use

Enter the radius (r) value
Enter the angle (θ) value
Select whether the angle is in degrees or radians
View the x and y coordinates and complex number form

Convert polar coordinates to rectangular (Cartesian) form

Convert polar coordinates (r, θ) to rectangular/Cartesian coordinates (x, y). Method: Convert the selected angle unit to radians, then multiply radius by cosine and sine to obtain Cartesian x and y. Negative radii are allowed by the coordinate convention.

A point is (r, θ), where r is a signed radial coordinate and θ is measured counterclockwise from the positive x-axis; distance from the origin is |r|.

Conversion Formulas

To convert from polar (r, θ) to rectangular (x, y) coordinates:

x = r × cos(θ)
y = r × sin(θ)

To convert from rectangular (x, y) to polar (r, θ) coordinates:

r = √(x² + y²)
θ = atan2(y, x) for the appropriate quadrant (reverse conversion)

Connection to Complex Numbers

Polar coordinates are closely related to complex numbers. A complex number z = x + yi can be written in polar form as z = r(cos θ + i sin θ) or using Euler's formula as z = re^(iθ).

This connection makes polar form especially useful for multiplication and division of complex numbers, as well as finding roots.

Common Angle Conversions

Degrees	Radians	cos(θ)	sin(θ)
0°	0	1	0
30°	π/6	√3/2	1/2
45°	π/4	√2/2	√2/2
60°	π/3	1/2	√3/2
90°	π/2	0	1
180°	π	-1	0
270°	3π/2	0	-1

Worked scenario

Point from radius and angle: r=2, θ=90° → (x,y)=(0,2).

Method, example, rounding, and limits

Method: Convert the selected angle unit to radians, then multiply radius by cosine and sine to obtain Cartesian x and y. Negative radii are allowed by the coordinate convention.

Worked example: r=2, θ=90° → (x,y)=(0,2).

Rounding: Cartesian coordinates and the radian angle use up to six decimal places; the degree angle uses two decimal places.

Limit: Rounding can make near-zero trigonometric components display as zero.

Reference: OpenStax. https://openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates

Frequently Asked Questions

When should I use polar coordinates instead of rectangular?: Polar coordinates are ideal when dealing with circular or rotational problems, such as describing orbits, spirals, or any situation where distance from a center point and angle are more natural measurements than x and y positions.
Can the radius be negative?: Yes, a negative radius means the point is in the opposite direction. The point (-r, θ) is the same as (r, θ + 180°). This convention is sometimes used in mathematics but can be confusing, so positive radii are more common.
How do I convert degrees to radians?: Multiply degrees by π/180. For example, 90° = 90 × π/180 = π/2 radians. Conversely, multiply radians by 180/π to get degrees.
What's the relationship between polar form and complex numbers?: A complex number x + yi corresponds to the point (x, y) in rectangular coordinates, which equals (r, θ) in polar form where r = √(x² + y²) and θ = atan2(y, x). This is written as r∠θ or r·e^(iθ).

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Math

Polar to Rectangular Calculator – Convert Coordinates

Method

Variables and units

Worked example

Limitations

Sources

How to Use

Convert polar coordinates to rectangular (Cartesian) form

Conversion Formulas

Connection to Complex Numbers

Common Angle Conversions

Worked scenario

Method, example, rounding, and limits

Frequently Asked Questions

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