Calculator Online

Home AI Chat AI Math Solver

NEW

Tools ▾ Category ▾

Pricing Sign In

Calculator Online

Your Result is copied!

Polar Coordinates Calculator

Make conversions between rectangular (Cartesian) and polar plane coordinates.

Related

Polar Coordinates Calculator

This calculator allows you to convert between Cartesian (rectangular) coordinates and polar coordinates in 2D space. It provides detailed step-by-step solutions for both conversions.

What Are Cartesian and Polar Coordinates?

Cartesian and Polar Coordinates

1. Polar Coordinates

Polar coordinates represent a point by its distance from the origin and an angle relative to the x-axis. Denoted as (r, θ):

r = distance from the origin to the point
θ = angle between the positive x-axis and the line to the point

2. Cartesian (Rectangular) Coordinates

Cartesian coordinates specify a point as (x, y) in the XY-plane:

x = horizontal distance from the origin (positive right, negative left)
y = vertical distance from the origin (positive up, negative down)

How to Convert Between Cartesian and Polar Coordinates

1. Manual Calculations

1.1 Cartesian → Polar

For a point (x, y), the polar coordinates (r, θ) are calculated as:

r = √(x² + y²)

θ = arctan(y / x)

Cartesian to Polar Conversion

1.2 Polar → Cartesian

For a point (r, θ), Cartesian coordinates (x, y) are:

x = r cos θ

y = r sin θ

Polar to Cartesian Conversion

2. Using the Polar Coordinates Calculator

Select the conversion type: Cartesian → Polar or Polar → Cartesian
Enter the required coordinate values
Click Calculate to obtain the result

Worked Examples

Example 1: Cartesian → Polar

Given point: (x, y) = (6, 2)

Step 1: Calculate r

r = √(6² + 2²) = √(36 + 4) = √40 ≈ 6.325

Step 2: Calculate θ

θ = arctan(2 / 6) = arctan(1/3) ≈ 18.43°

Example 2: Polar → Cartesian

Given point: (r, θ) = (2, 45°)

Step 1: Calculate x

x = 2 × cos(45°) ≈ 2 × 0.707 ≈ 1.414

Step 2: Calculate y

y = 2 × sin(45°) ≈ 2 × 0.707 ≈ 1.414

Sample Cartesian ↔ Polar Conversions

Cartesian (x, y)	Polar (r, θ)
(3, 4)	5, arctan(4/3) ≈ 53.13°
(-2, 1)	√5, arctan(1 / -2) ≈ -26.57°
(0, 5)	5, 90°
(-4, 4)	√32, arctan(4 / -4) ≈ 135°

FAQs

Are polar coordinates unique?

No. A point can have infinitely many polar representations since the angle θ can differ by multiples of 2π.

How is z represented in polar form?

For complex numbers z = x + iy:

z = r e^(iθ), where:

x = real part
y = imaginary part

What is the polar form of 1?

1 in polar form is:

1 = 1 e^(i 0) = 1 e^(i 2π), as sine and cosine are periodic with period 2π.

References

Wikipedia: Polar coordinate system
Math Insight: Conversion formulas and Cartesian plane

Related

Calculator Online

Get the ease of calculating anything from the source of calculator online

Links

Home Conversion Calculator About Calculator Online Blog Hire Us Knowledge Base Sitemap Sitemap Two

Support

Email us at