We are given the equation for the surface area \( A \) of a cylinder:

\[
A = 2\pi r(r + h)
\]

We are asked to solve for \( h \), the height of the cylinder. Here's how we proceed:

### Step 1: Expand the equation
First, distribute \( 2\pi r \) to both terms inside the parentheses:

\[
A = 2\pi r^2 + 2\pi r h
\]

### Step 2: Isolate the term with \( h \)
Subtract \( 2\pi r^2 \) from both sides:

\[
A - 2\pi r^2 = 2\pi r h
\]

### Step 3: Solve for \( h \)
Now, divide both sides by \( 2\pi r \) to isolate \( h \):

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

Thus, the solution for \( h \) is:

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

Would you like further details or any clarifications?

### Relative Questions:
1. What does the term \( 2\pi r(r+h) \) represent geometrically in a cylinder?
2. How does changing the radius \( r \) affect the value of \( h \)?
3. Can you apply this equation if the cylinder is not a right cylinder?
4. What happens to the height \( h \) if the surface area \( A \) increases while keeping \( r \) constant?
5. How would you modify the equation if you're solving for the radius \( r \)?

### Tip:
When solving for a variable, always ensure that terms involving the unknown variable are grouped together before isolating it.

Learn how to solve for the height h in the surface area formula of a cylinder, A = 2πr(r + h). This step-by-step solution covers important algebraic and geometric concepts, ideal for students in Grades 8-10.

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