Sphere Surface Area to Diameter Calculation

The question asks us to find the diameter of a sphere when its surface area is given as $400\pi$ mm$^2$. We need to use the formula for the surface area of a sphere and the relationship between radius and diameter.

Sphere Formulas

The key formulas needed are:

• Surface Area of a sphere ($A$): $A = 4\pi r^2$, where $r$ is the radius.
• Diameter of a sphere ($d$): $d = 2r$.

Calculating the Sphere's Radius

We are given the surface area $A = 400\pi$ mm$^2$. We can use the surface area formula to find the radius ($r$).

1. Start with the surface area formula:

   $A = 4\pi r^2$

2. Substitute the given surface area value into the formula:

   $400\pi = 4\pi r^2$

3. To solve for $r^2$, divide both sides of the equation by $4\pi$:

   $r^2 = \frac{400\pi}{4\pi}$

   $r^2 = 100$

4. Now, find the radius ($r$) by taking the square root of $r^2$:

   $r = \sqrt{100}$

   $r = 10$ mm

Determining the Sphere's Diameter

Once we have the radius, we can easily calculate the diameter using the formula $d = 2r$.

1. Use the diameter formula:

   $d = 2r$

2. Substitute the calculated radius ($r = 10$ mm):

   $d = 2 \times 10$ mm

   $d = 20$ mm

So, the diameter of the sphere is 20 mm. This matches the second option provided.