Calculate the sphericity of a cube.

Steps to Solve

1. Understand the formula for sphericity

Sphericity $\Psi$ is defined as the ratio of the surface area of a sphere with the same volume as the given shape to the surface area of the shape itself. The formula is given by:

$$ \Psi = \frac{(36\pi V^2)}{A^3} $$

where $V$ is the volume of the shape and $A$ is the surface area.

2. Calculate the volume of the cube

For a cube with side length $s$, the volume $V$ is calculated as:

$$ V = s^3 $$

3. Calculate the surface area of the cube

The surface area $A$ of a cube is calculated as:

$$ A = 6s^2 $$

4. Substitute the volume and surface area into the sphericity formula

Now substitute the expressions for $V$ and $A$ into the sphericity formula:

$$ \Psi = \frac{(36\pi (s^3)^2)}{(6s^2)^3} $$

5. Simplify the formula

Now simplify the expression step-by-step:

First, calculate $V^2$ and $A^3$:

• $V^2 = (s^3)^2 = s^6$

• $A^3 = (6s^2)^3 = 216s^6$

Now substitute these into the sphericity formula:

$$ \Psi = \frac{36\pi s^6}{216s^6} $$

6. Final simplification

Now simplify the fraction:

$$ \Psi = \frac{36\pi}{216} = \frac{\pi}{6} $$