Solve for r M=pir^2hd ; for r

$M = π r^{2} h d$ ; for $r$

Rewrite the equation as $π r^{2} h d = M$ .

$π r^{2} h d = M$

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

Divide each term in $π r^{2} h d = M$ by $π h d$ .

$\frac{π r^{2} h d}{π h d} = \frac{M}{π h d}$

Simplify the left side.

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π r^{2} h d}{π h d} = \frac{M}{π h d}$

Rewrite the expression.

$\frac{r^{2} h d}{h d} = \frac{M}{π h d}$

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{r^{2} h d}{h d} = \frac{M}{π h d}$

Rewrite the expression.

$\frac{r^{2} d}{d} = \frac{M}{π h d}$

Cancel the common factor of $d$ .

Cancel the common factor.

$\frac{r^{2} d}{d} = \frac{M}{π h d}$

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{M}{π h d}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{\frac{M}{π h d}}$

Simplify $\pm \sqrt{\frac{M}{π h d}}$ .

Rewrite $\sqrt{\frac{M}{π h d}}$ as $\frac{\sqrt{M}}{\sqrt{π h d}}$ .

$r = \pm \frac{\sqrt{M}}{\sqrt{π h d}}$

Multiply $\frac{\sqrt{M}}{\sqrt{π h d}}$ by $\frac{\sqrt{π h d}}{\sqrt{π h d}}$ .

$r = \pm \frac{\sqrt{M}}{\sqrt{π h d}} \cdot \frac{\sqrt{π h d}}{\sqrt{π h d}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{M}}{\sqrt{π h d}}$ by $\frac{\sqrt{π h d}}{\sqrt{π h d}}$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{\sqrt{π h d} \sqrt{π h d}}$

Raise $\sqrt{π h d}$ to the power of $1$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{\sqrt{π h d}}^{1} \sqrt{π h d}}$

Raise $\sqrt{π h d}$ to the power of $1$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{\sqrt{π h d}}^{1} {\sqrt{π h d}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{\sqrt{π h d}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{\sqrt{π h d}}^{2}}$

Rewrite ${\sqrt{π h d}}^{2}$ as $π h d$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{π h d}$ as ${(π h d)}^{\frac{1}{2}}$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{({(π h d)}^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{(π h d)}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{(π h d)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{(π h d)}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{{(π h d)}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{M} \sqrt{π h d}}{π h d}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{M π h d}}{π h d}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{M π h d}}{π h d}$

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{M π h d}}{π h d}$

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{M π h d}}{π h d}$

$r = - \frac{\sqrt{M π h d}}{π h d}$

$r = \frac{\sqrt{M π h d}}{π h d}$

$r = - \frac{\sqrt{M π h d}}{π h d}$

Solve for r M=pir^2hd ; for r

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