Degrees to Polar Calculator

Convert degrees to polar coordinates with our free degrees to polar calculator. Polar coordinates represent points in a plane using a distance from a reference point and an angle from a reference direction. This calculator helps you convert angle measurements from degrees to polar coordinates.

What are Polar Coordinates?

Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Unlike Cartesian coordinates (x, y), which specify a point's horizontal and vertical distances from the origin, polar coordinates use a radius (r) and an angle (θ).

The reference point is typically the origin (0,0) of the coordinate system, and the reference direction is usually along the positive x-axis. The angle θ is measured in radians or degrees from this reference direction, with positive angles indicating counterclockwise rotation and negative angles indicating clockwise rotation.

Key Properties of Polar Coordinates

Polar coordinates are useful for describing circular or spiral patterns.
They simplify calculations involving rotation and symmetry.
Polar coordinates are commonly used in physics, engineering, and computer graphics.

How to Convert Degrees to Polar Coordinates

Converting degrees to polar coordinates involves calculating the radius (r) and angle (θ) from a given point in Cartesian coordinates (x, y). The process involves using trigonometric functions to determine the angle and the distance from the origin.

Step-by-Step Conversion Process

Identify the Cartesian coordinates (x, y) of the point you want to convert.
Calculate the radius (r) using the Pythagorean theorem: r = √(x² + y²).
Calculate the angle (θ) in degrees using the arctangent function: θ = arctan(y/x).
Adjust the angle based on the quadrant in which the point lies.
Express the result in polar coordinates as (r, θ).

Conversion Formulas

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

r = √(x² + y²)
θ = arctan(y/x) (in degrees)

Formula for Conversion

The formula for converting degrees to polar coordinates involves calculating the radius (r) and angle (θ) from Cartesian coordinates (x, y). The radius is calculated using the Pythagorean theorem, and the angle is calculated using the arctangent function.

Polar Coordinates Formula

Given Cartesian coordinates (x, y), the polar coordinates (r, θ) are calculated as follows:

r = √(x² + y²)
θ = arctan(y/x) (in degrees)

This formula allows you to convert any point from Cartesian coordinates to polar coordinates. The radius (r) represents the distance from the origin, and the angle (θ) represents the direction from the positive x-axis.

Worked Example

Let's convert the Cartesian coordinates (3, 4) to polar coordinates using the formula.

Calculate the radius (r): r = √(3² + 4²) = √(9 + 16) = √25 = 5.
Calculate the angle (θ): θ = arctan(4/3) ≈ 53.13°.

The polar coordinates for the point (3, 4) are (5, 53.13°).

Verification

To verify the conversion, you can convert the polar coordinates back to Cartesian coordinates using the formulas:

x = r * cos(θ)
y = r * sin(θ)

For (5, 53.13°):

x ≈ 5 * cos(53.13°) ≈ 3
y ≈ 5 * sin(53.13°) ≈ 4

This confirms the conversion is correct.

FAQ

What are polar coordinates used for?

Polar coordinates are used in various fields such as physics, engineering, and computer graphics to describe points in a plane using a distance from a reference point and an angle from a reference direction.

How do I convert degrees to polar coordinates?

To convert degrees to polar coordinates, you need to calculate the radius (r) using the Pythagorean theorem and the angle (θ) using the arctangent function. The formula for conversion is r = √(x² + y²) and θ = arctan(y/x) in degrees.

What is the difference between polar and Cartesian coordinates?

Polar coordinates represent points using a distance from a reference point and an angle from a reference direction, while Cartesian coordinates represent points using horizontal and vertical distances from the origin.