Spherical Coordinates Calculator

In a three-dimensional space, a point can be defined by its Cartesian coordinates (x, y, z) or by its spherical coordinates (r, theta, phi). To convert from Cartesian to spherical coordinates, the calculator uses the radial distance formula r = sqrt(x^2 + y^2 + z^2), the azimuthal angle phi = atan2(y, x), and the polar angle theta = acos(z / r). Conversely, to convert spherical coordinates back to Cartesian form, the system applies the trigonometric equations x = r * sin(theta) * cos(phi), y = r * sin(theta) * sin(phi), and z = r * cos(theta).

Let us convert a point given in Cartesian coordinates where x = 3, y = 4, and z = 12 into spherical coordinates. First, calculate the radial distance r: r = sqrt(3^2 + 4^2 + 12^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13. Next, determine the azimuthal angle phi using the arctangent of y over x: phi = atan2(4, 3) which is approximately 0.927 radians or 53.13 degrees. Finally, calculate the polar angle theta by taking the arccosine of z divided by r: theta = acos(12 / 13), yielding approximately 0.395 radians or 22.62 degrees. Thus, the spherical coordinate representation is (13, 0.927, 0.395).

When working with spherical coordinates, always double-check whether your output angles are expected in degrees or radians to prevent calculation mismatches in downstream equations. Pay close attention to the quadrant of your x and y values when calculating the azimuthal angle, as standard arctangent functions can occasionally yield ambiguous signs if the two-argument atan2 function is not utilized properly.