convert spherical to cylindrical coordinates

Step by step worksheet, Spherical to Cylindrical

Converts **Spherical to Cylindrical **\[r=\rho\sin\phi \, \ \theta = \theta , \ z=\rho\cos\phi\]

Converting spherical to cylindrical coordinates calculator converts spherical coordinates (ρ, θ, φ) to cylindrical coordinates (r, θ, z)

Spherical coordinates are a coordinate system used to describe the position of a point in three-dimensional space. They are especially useful when dealing with problems involving objects that have a spherical symmetry, such as planets or spheres.

In the spherical coordinate system, a point is described using three values: the radial distance (r), the polar angle (θ), and the azimuthal angle (φ).

**Radial Distance (r):**

The radial distance is the distance from the origin (usually denoted as O) to the point. It is measured along the straight line connecting the origin to the point.**Polar Angle (θ):**

The polar angle is the angle between the positive z-axis and the line connecting the origin to the point. It is measured in the range of 0° to 180° (or 0 to π radians). The polar angle is also known as the inclination angle or the zenith angle.**Azimuthal Angle (φ):**

The azimuthal angle is the angle formed between the positive x-axis and the projection of the line connecting the origin to the point onto the xy-plane. It is measured in the range of 0° to 360° (or 0 to 2π radians). The azimuthal angle is also known as the azimuth angle or the azimuthal coordinate.

To convert spherical coordinates (**ρ, θ, φ**) to cylindrical coordinates (**r, θ, z**), you can use the following equations

1. Express the radial distance (r) in terms of the cylindrical coordinate ρ:

**r = ρ sin(θ)**

2. Express the azimuthal angle (φ) in terms of the cylindrical coordinate θ:

**φ = θ**

3. Determine the value of z using the polar angle (θ), as follows:

**z = r cos(θ)****= (ρ sin(θ)) cos(θ)****= ρ sin(θ) cos(θ)**

So, the conversion formulas for spherical coordinates to cylindrical coordinates are:

**ρ = r sin(φ)****θ = φ****z = r cos(φ)**

It's important to note that the ranges of the coordinates vary depending on the specific system used. In the case of cylindrical coordinates, ρ is typically non-negative (ρ ≥ 0), θ is measured in the range of 0 to 2π radians, and z has no restrictions.

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