# Solve for h: V=1/3 pi r^2 h?

Sep 7, 2017

Move terms that don't have $h$ in them to the other side (using addition/subtraction), then move all factors other than $h$ to the other side (using multiplication/division).

#### Explanation:

If we want to solve $V = \frac{1}{3} \pi {r}^{2} h$ for $h$, we need to isolate the term with $h$ (already done), and then multiply both sides by the inverses of everything other than $h$.

$\textcolor{w h i t e}{\implies} V \textcolor{w h i t e}{\times \frac{3}{\pi {r}^{2}}} = \frac{1}{3} \pi {r}^{2} h$

$\implies V \textcolor{red}{\times \frac{3}{\pi {r}^{2}}} = \frac{1}{3} \pi {r}^{2} h \textcolor{red}{\times \frac{3}{\pi {r}^{2}}}$

The multiplication by $\frac{3}{\pi {r}^{2}}$ to both sides is our choice; we do this so that each piece other than $h$ has a multiplicative inverse that cancels it off.

$\implies V \times \frac{3}{\pi {r}^{2}} = \frac{1}{\textcolor{\mathmr{and} a n \ge}{\cancel{\textcolor{b l a c k}{3}}}} \textcolor{m a \ge n t a}{\cancel{\textcolor{b l a c k}{\pi {r}^{2}}}} h \times \frac{\textcolor{\mathmr{and} a n \ge}{\cancel{\textcolor{b l a c k}{3}}}}{\textcolor{m a \ge n t a}{\cancel{\textcolor{b l a c k}{\pi {r}^{2}}}}}$

$\implies \textcolor{w h i t e}{V \times} \frac{3 V}{\pi {r}^{2}} = \textcolor{w h i t e}{\frac{1}{\cancel{3}} \cancel{\pi {r}^{2}}} h$

Thus, our cone volume formula, when solved for $h$, is

$h = \frac{3 V}{\pi {r}^{2}}$
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