Solve for h S=pir^2+2pirh

$S = π r^{2} + 2 π r h$

Rewrite the equation as $π r^{2} + 2 π r h = S$ .

$π r^{2} + 2 π r h = S$

Subtract $π r^{2}$ from both sides of the equation.

$2 π r h = S - π r^{2}$

Divide each term in $2 π r h = S - π r^{2}$ by $2 π r$ and simplify.

Divide each term in $2 π r h = S - π r^{2}$ by $2 π r$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $π$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $r$ .

Cancel the common factor.

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

Divide $h$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $π$ .

Cancel the common factor.

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

Rewrite the expression.

$h = \frac{S}{2 π r} + \frac{- 1 r^{2}}{2 r}$

Cancel the common factor of $r^{2}$ and $r$ .

Factor $r$ out of $- 1 r^{2}$ .

$h = \frac{S}{2 π r} + \frac{r (- 1 r)}{2 r}$

Cancel the common factors.

Factor $r$ out of $2 r$ .

$h = \frac{S}{2 π r} + \frac{r (- 1 r)}{r \cdot 2}$

Cancel the common factor.

$h = \frac{S}{2 π r} + \frac{r (- 1 r)}{r \cdot 2}$

Rewrite the expression.

$h = \frac{S}{2 π r} + \frac{- 1 r}{2}$

Move the negative in front of the fraction.

$h = \frac{S}{2 π r} - \frac{r}{2}$