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To solve the formula for \( h \) from the surface area equation of a cylinder, let's start by analyzing the given formula:

\[
S = 2\pi r^2 + 2\pi rh
\]

Here, \( S \) is the surface area, \( r \) is the radius of the cylinder's base, and \( h \) is the height of the cylinder. We need to solve this equation for \( h \).

### Steps to solve for \( h \):

1. **Isolate the term involving \( h \):**

   Subtract \( 2\pi r^2 \) from both sides of the equation to isolate the term with \( h \):

   \[
   S - 2\pi r^2 = 2\pi rh
   \]

2. **Solve for \( h \):**

   Now, divide both sides of the equation by \( 2\pi r \) to solve for \( h \):

   \[
   h = \frac{S - 2\pi r^2}{2\pi r}
   \]

So, the final expression for \( h \) is:

\[
h = \frac{S - 2\pi r^2}{2\pi r}
\]

This is the height of the cylinder expressed in terms of the surface area \( S \) and the radius \( r \).

Would you like further details on how this formula is derived or how it can be applied? 

Here are some related questions:
1. What is the formula for the volume of a cylinder, and how does it relate to the surface area?
2. How can you derive the surface area of a cylinder starting from its components?
3. What happens to the height \( h \) when the surface area \( S \) is doubled?
4. How does changing the radius \( r \) affect the height \( h \) for a fixed surface area?
5. Can this formula for \( h \) be applied to other types of cylinders, such as oblique cylinders?

**Tip:** When manipulating formulas, always perform the same operation on both sides of the equation to maintain equality.

Solve the formula for h from the surface area equation of a cylinder: S = 2πr² + 2πrh

Learn how to solve for the height (h) in the surface area equation of a cylinder. This problem demonstrates algebraic manipulation and the geometric understanding of a cylinder's surface area. Suitable for students in Grades 8-10.

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