Find The Polar Coordinates Calculator

This Polar Coordinates Calculator helps you convert Cartesian coordinates (x, y) into their polar form (r, θ), providing the radial distance (r) and the angle (θ) in both radians and degrees. Enter your x and y values to get the polar coordinates instantly.

Cartesian to Polar Converter

Intermediate Values Table

Input/Value	Value
x	3
y	4
x²	9
y²	16
x² + y²	25
r = √(x² + y²)	5
θ (radians) = atan2(y, x)	0.927
θ (degrees)	53.130

Table showing input values and calculated intermediate and final polar coordinates.

What is a Polar Coordinates Calculator?

A Polar Coordinates Calculator is a tool used to convert coordinates from the Cartesian (or rectangular) coordinate system (x, y) to the polar coordinate system (r, θ). In the Cartesian system, a point is located using its horizontal (x) and vertical (y) distances from the origin. In the polar system, a point is located by its distance (r) from the origin (pole) and the angle (θ) its line segment to the origin makes with the positive x-axis (polar axis).

This calculator is useful for students, engineers, physicists, and anyone working with geometry or vector analysis where polar coordinates simplify calculations or representations. For example, describing circular or spiral motion is often easier in polar coordinates.

Who should use it?

• Mathematics Students: Learning about coordinate systems and transformations.
• Physics Students and Professionals: Analyzing motion, fields, or waves with radial symmetry.
• Engineers: Designing systems or analyzing forces where angles and distances from a central point are key.
• Navigators and Astronomers: Plotting positions based on distance and direction.

Common Misconceptions

A common misconception is that the angle θ is always just atan(y/x). While this is related, using atan2(y, x) is crucial because it considers the signs of both x and y to place the angle in the correct quadrant (0 to 360° or -180° to 180°), whereas atan(y/x) only returns values between -90° and 90°.

Polar Coordinates Calculator Formula and Mathematical Explanation

To convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), we use the following formulas based on the Pythagorean theorem and trigonometry:

1. Calculate the radius (r): The distance 'r' from the origin (0,0) to the point (x,y) is found using the distance formula, which is derived from the Pythagorean theorem:
   r = √(x² + y²)
2. Calculate the angle (θ): The angle 'θ' is the angle between the positive x-axis and the line segment from the origin to the point (x,y). It is calculated using the atan2(y, x) function, which correctly determines the quadrant of the angle:
   θ (in radians) = atan2(y, x)
   The atan2(y, x) function is preferred over atan(y/x) because it handles cases where x=0 and gives the angle in the full range (-π to π or -180° to 180°), correctly placing it in one of the four quadrants based on the signs of x and y.
3. Convert angle to degrees (optional): If the angle is needed in degrees, convert from radians:
   θ (in degrees) = θ (in radians) × (180 / π)

Variables Table

Variable	Meaning	Unit	Typical Range
x	Cartesian x-coordinate	Length units	-∞ to +∞
y	Cartesian y-coordinate	Length units	-∞ to +∞
r	Polar radius (magnitude/distance from origin)	Length units	0 to +∞
θ	Polar angle (direction from positive x-axis)	Radians or Degrees	-π to π rad, or -180° to 180° (or 0 to 2π, 0° to 360°)

Practical Examples (Real-World Use Cases)

Example 1: Point (3, 4)

Let's convert the Cartesian coordinates (x=3, y=4) to polar coordinates.

• r = √(3² + 4²) = √(9 + 16) = √25 = 5
• θ = atan2(4, 3) ≈ 0.927 radians
• θ ≈ 0.927 × (180 / π) ≈ 53.13 degrees

So, the polar coordinates are (r=5, θ≈53.13°).

Example 2: Point (-2, -2)

Let's convert the Cartesian coordinates (x=-2, y=-2) to polar coordinates.

• r = √((-2)² + (-2)²) = √(4 + 4) = √8 ≈ 2.828
• θ = atan2(-2, -2) ≈ -2.356 radians
• θ ≈ -2.356 × (180 / π) ≈ -135 degrees (or 225 degrees)

So, the polar coordinates are (r≈2.828, θ=-135° or 225°). Our Polar Coordinates Calculator will typically give the angle between -180° and 180° or 0 and 360° based on the `atan2` implementation.

How to Use This Polar Coordinates Calculator

1. Enter X-coordinate: Input the value of the x-coordinate into the "X-coordinate (x)" field.
2. Enter Y-coordinate: Input the value of the y-coordinate into the "Y-coordinate (y)" field.
3. View Results: The calculator will automatically update and display:
   • The primary result showing (r, θ) in degrees.
   • The radius (r).
   • The angle (θ) in both radians and degrees.
   • Intermediate values like x², y², and x²+y².
4. Visualize: The chart below the calculator shows the point (x,y), the vector from the origin, and the angle θ.
5. Reset: Click "Reset" to clear the fields and return to default values.
6. Copy: Click "Copy Results" to copy the main results and inputs to your clipboard.

The Polar Coordinates Calculator provides immediate feedback as you type.

Key Factors That Affect Polar Coordinates Results

1. Value of x: The horizontal position significantly affects both r and θ.
2. Value of y: The vertical position also significantly affects both r and θ.
3. Signs of x and y: The signs determine the quadrant of the angle θ, which is correctly handled by atan2(y, x).
4. Magnitude of x and y: Larger magnitudes of x or y lead to a larger radius r.
5. Ratio y/x: The ratio influences the angle θ, but atan2 uses both values independently to determine the correct quadrant.
6. Origin (0,0): If both x and y are 0, r is 0, and θ is undefined or conventionally 0. Our Polar Coordinates Calculator handles this.

Frequently Asked Questions (FAQ)

What if x is 0?

If x=0 and y>0, θ is π/2 radians (90°). If x=0 and y<0, θ is -π/2 radians (-90° or 270°). If x=0 and y=0, r=0 and θ is undefined (often taken as 0). The Polar Coordinates Calculator uses `atan2` which handles this.

What if y is 0?

If y=0 and x>0, θ is 0 radians (0°). If y=0 and x<0, θ is π radians (180°). If x=0 and y=0, r=0 and θ is undefined.

What if both x and y are 0?

If x=0 and y=0, the point is at the origin. The radius r is 0, and the angle θ is undefined, although it is often taken to be 0 for convenience.

What is the range of the angle θ?

The `atan2` function typically returns an angle between -π and π radians (-180° and 180°). You can add 2π (or 360°) to negative angles to get them in the 0 to 2π (or 0° to 360°) range if needed.

Can r be negative?

In standard polar coordinates, r is defined as the non-negative distance from the origin (r ≥ 0). However, some contexts allow negative r, where (-r, θ) represents (r, θ + π).

Why use atan2(y, x) instead of atan(y/x)?

atan(y/x) only gives angles between -π/2 and π/2 (-90° and 90°), so you'd need to adjust for the correct quadrant based on the signs of x and y. atan2(y, x) does this automatically.

Are the units of x and y important?

Yes, the unit of r will be the same as the units of x and y. Ensure x and y are in the same units.

How accurate is this Polar Coordinates Calculator?

The calculations are based on standard mathematical formulas and JavaScript's `Math` functions, providing high precision. The display is rounded for readability.

Related Tools and Internal Resources

• {related_keywords[0]}: Our main tool for converting Cartesian to Polar coordinates.
• {related_keywords[5]}: Calculate the magnitude of a vector given its components, similar to finding 'r'.
• {related_keywords[3]}: Find the angle between two vectors, related to the angle in polar coordinates.
• {related_keywords[4]}: Understand the Pythagorean theorem used to find 'r'.
• {related_keywords[4]}: Explore trigonometric functions like atan2.
• Unit Circle Calculator: Visualize angles and trigonometric values, relevant to understanding θ.