Solve the cylinder surface area formula S = 2 pi r h + 2 pi r^2 for h.

Step-by-step solution

1. 1

   See what the formula is made of. $S$ is the total surface area of a cylinder: $2\pi r h$ is the curved side (a rolled-out rectangle) and $2\pi r^2$ is the two circular ends. Only the first term contains $h$, so the second term has to be removed before dividing.

2. 2

   Undo the addition first. Subtract $2\pi r^2$ from both sides:

   $S - 2\pi r^2 = 2\pi r h$

3. 3

   Divide by the whole coefficient of $h$. The coefficient is $2\pi r$, so divide both sides by it:

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

4. 4

   Split the fraction to get the tidy form. Dividing each term of the numerator separately:

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

   So the two forms $\dfrac{S-2\pi r^2}{2\pi r}$ and $\dfrac{S}{2\pi r} - r$ are the same answer.

5. 5

   Reject the near-miss options. $\dfrac{S - r}{2\pi r}$ subtracts $r$ instead of $2\pi r^2$; $S - \dfrac{2\pi}{r}$ and $S - \dfrac{r}{2\pi}$ never divide by the coefficient at all. Any correct answer must leave $h$ alone with no $h$ on the right.

6. 6

   Check with numbers. Take $r = 3$ and $h = 5$: $S = 2\pi(3)(5) + 2\pi(9) = 30\pi + 18\pi = 48\pi$. Now run the formula backwards: $\dfrac{48\pi}{2\pi(3)} - 3 = 8 - 3 = 5$ ✓, recovering the original height.