Cartesian to Spherical Coordinates calculator

$\color{red} \bold{Related \space Calculators:}$
Polar to Cartesian coordinates
Cartesian to Polar coordinates
Cartesian to Cylindrical coordinates
Cylindrical to Cartesian coordinates
Cylindrical to Spherical coordinates
Spherical to Cartesian coordinates
Spherical to Cylindrical coordinates

$\bold{Table \space of \space Content}$

• 1. Introduction to the Cartesian to Spherical coordinates calculator
• 2. What is the Formulae used ?
• 3. How do I convert the Cartesian coordinates to Spherical?
• 4. Why choose our Cartesian to Spherical coordinates Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Cartesian to Spherical coordinates calculator: -

Understanding different coordinate systems is crucial in various fields, particularly three-dimensional space. This blog post will delve into converting Cartesian coordinates to spherical coordinates, clarifying the definitions, formulas, and real-world applications of this fundamental transformation.
$\bold{What \space are \space Cartesian \space Coordinates?}$
Cartesian coordinates locate points in the argand plane using a pair of perpendicular axes (usually denoted as x,y, and z) intersecting at the origin (0, 0, 0). A Point is represented as (x, y, Z), where x represents the x coordinate or horizontal distance and y & z represents the y & z coordinate or vertical distance of the point from the origin. We also call it a rectangular coordinate system.
Here, x, y, and z can attain any real value.
$\bold{What \space are \space Spherical \space Coordinates?}$
Spherical coordinates, on the other hand, represent a point using its radial distance (r), polar angle (θ), and azimuthal angle (ϕ).

2. What is the Formulae used?

To convert Cartesian coordinates (x,y,z) to spherical coordinates (r,θ,ϕ), the following formulas are used:
r = $\sqrt{(x^2 + y^2 + z^2)}$
$\theta$ = tan$^{-1}(\frac{y}{x})$
$\phi$ = tan$^{-1}(\frac{\sqrt{(x^2 + y^2)}}{z})$

3. How do I convert the Cartesian coordinates to Spherical?

Identify the given cartesian coordinates (x, y, z) of the point.
Calculate the radial distance or magnitude of the point from the origin by using the formula r = $\sqrt{x^2 + y^2 + z^2}.$
Determine the quadrant in which the point is lying.
Now determine the angle θ by using tan inverse formula.
Convert the angle to the desired units (degrees or radians), as the question asks.
Now determine the angle ϕ by using tan inverse formula.
Convert the angle to the desired units (degrees or radians), as the question asks.
Write down the point's Spherical coordinates (r, θ, ϕ).

4. Why choose our Cartesian to Spherical coordinates Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on how to convert the Cartesian coordinates to Spherical.

6. How to use this calculator

This calculator will help you convert the cartesian coordinates to spherical ones.
In the input boxes, you must put the values x, y, and z.
After clicking the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question: 1}$
Convert the given Cartesian coordinates into Spherical coordinates (1, 1, 1).
$\bold{Solution:1}$
r = $\sqrt{(1)^2 + (1)^2 + (1)^2}$ = $\sqrt{3}$
Since x > 0 & y > 0, the point lies in the 1st quadrant.
Polar angle = $\theta = tan^{-1}(\frac{1}{1})$ = 45 degrees or $\frac{\pi}{4}$ radian
Since z > 0, then (Point lies in the upper half segment above xy - plane toward Positive z-axis)
Azimuthal angle = $\phi$ = tan$^{-1}(\frac{\sqrt{2}}{1})$ = 55 degrees or $\frac{11\pi}{36}$ radian

$\bold{Question: 2}$
Convert the given Cartesian coordinates as (1, -1, -1) into Spherical coordinates.
$\bold{Solution:2}$
r = $\sqrt{(1)^2 + (-1)^2 + (-1)^2}$ = $\sqrt{3}$
Since x > 0 & y < 0, the point lies in the 4th quadrant.
Polar angle = $\theta = 2\pi + tan^{-1}(\frac{-1}{1})$ = 315 degrees or $\frac{7\pi}{4}$ radian
Since z < 0, then (Point lies in the lower half segment below xy - plane toward negative z-axis)
Azimuthal angle = $\phi$ = $2\pi$ + tan$^{-1}(\frac{\sqrt{2}}{-1})$ = 125 degrees or $\frac{25\pi}{36}$ radian

8. Frequently Asked Questions (FAQs)

Why use spherical coordinates?

Spherical coordinates are especially useful in physics and engineering, simplifying equations in systems with spherical symmetry.

What is the range of θ and ϕ?

θ ranges from 0 to π, and ϕ from 0 to 2π.

Can Cartesian coordinates be negative?

Yes, Cartesian coordinates can be negative, representing points in different octants of the three-dimensional space.

How do we convert back from spherical to Cartesian?

Use the formulas: x = rsin(θ)cos(ϕ), y = rsin(θ)sin(ϕ), z = rcos(θ).

What if the Cartesian coordinates are on the negative axes?

Adjust the signs in the formulas, accounting for the specific octant.

9. What are the real-life applications?

Spherical coordinates find applications in various fields, from astronomy for celestial navigation to physics simulations. In robotics, these coordinates aid in positioning and orientation, allowing efficient spatial representation.

10. Conclusion

Converting Cartesian coordinates to spherical coordinates is a fundamental skill with diverse applications. By mastering the formulas and understanding the nuances of this transformation, individuals can navigate and interpret spatial data more effectively in a wide range of scientific and engineering disciplines.

10. Conclusion

Converting Cartesian coordinates to spherical coordinates is a fundamental skill with diverse applications. By mastering the formulas and understanding the nuances of this transformation, individuals can navigate and interpret spatial data more effectively in a wide range of scientific and engineering disciplines.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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