Rearranging Formula Worksheets - Free Printable

Let's solve each of these formula rearrangement problems step by step. The goal is to isolate the variable we're solving for on one side of the equation.

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1) Solve $ V = \frac{1}{3} \pi r^2 h $ for $ h $

We want to solve for $ h $.
Start by multiplying both sides by 3 to eliminate the fraction:

$$
3V = \pi r^2 h
$$

Now divide both sides by $ \pi r^2 $:

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

✔ Answer: $ h = \frac{3V}{\pi r^2} $

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2) Solve $ F = ma $ for $ a $

Divide both sides by $ m $:

$$
a = \frac{F}{m}
$$

✔ Answer: $ a = \frac{F}{m} $

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3) Solve $ a = \frac{v - u}{t} $ for $ v $

Multiply both sides by $ t $:

$$
at = v - u
$$

Add $ u $ to both sides:

$$
v = at + u
$$

✔ Answer: $ v = at + u $

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4) Solve $ c^2 = a^2 + b^2 $ for $ b $

Subtract $ a^2 $ from both sides:

$$
b^2 = c^2 - a^2
$$

Take the square root of both sides:

$$
b = \sqrt{c^2 - a^2}
$$

(Note: We usually take the positive root unless context allows negative.)

✔ Answer: $ b = \sqrt{c^2 - a^2} $

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5) Solve $ A = \frac{1}{2}(a + b)h $ for $ b $

Multiply both sides by 2:

$$
2A = (a + b)h
$$

Divide both sides by $ h $:

$$
\frac{2A}{h} = a + b
$$

Subtract $ a $ from both sides:

$$
b = \frac{2A}{h} - a
$$

✔ Answer: $ b = \frac{2A}{h} - a $

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6) Solve $ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} $ for $ u $

Isolate $ \frac{1}{u} $:

$$
\frac{1}{u} = \frac{1}{f} - \frac{1}{v}
$$

Find a common denominator on the right-hand side:

$$
\frac{1}{u} = \frac{v - f}{fv}
$$

Now take the reciprocal of both sides:

$$
u = \frac{fv}{v - f}
$$

✔ Answer: $ u = \frac{fv}{v - f} $

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7) Solve $ K = \frac{1}{2}mv^2 $ for $ v $

Multiply both sides by 2:

$$
2K = mv^2
$$

Divide both sides by $ m $:

$$
v^2 = \frac{2K}{m}
$$

Take the square root:

$$
v = \sqrt{\frac{2K}{m}}
$$

✔ Answer: $ v = \sqrt{\frac{2K}{m}} $

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8) Solve $ \frac{x}{a} + \frac{y}{b} = 1 $ for $ x $

Subtract $ \frac{y}{b} $ from both sides:

$$
\frac{x}{a} = 1 - \frac{y}{b}
$$

Multiply both sides by $ a $:

$$
x = a\left(1 - \frac{y}{b}\right)
$$

Simplify:

$$
x = a - \frac{ay}{b}
$$

✔ Answer: $ x = a - \frac{ay}{b} $

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9) Solve $ y = mx + c $ for $ m $

Subtract $ c $ from both sides:

$$
y - c = mx
$$

Divide both sides by $ x $ (assuming $ x \neq 0 $):

$$
m = \frac{y - c}{x}
$$

✔ Answer: $ m = \frac{y - c}{x} $

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10) Solve $ P = I^2 R $ for $ R $

Divide both sides by $ I^2 $:

$$
R = \frac{P}{I^2}
$$

✔ Answer: $ R = \frac{P}{I^2} $

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✔ Final Answers Summary:

1) $ h = \frac{3V}{\pi r^2} $
2) $ a = \frac{F}{m} $
3) $ v = at + u $
4) $ b = \sqrt{c^2 - a^2} $
5) $ b = \frac{2A}{h} - a $
6) $ u = \frac{fv}{v - f} $
7) $ v = \sqrt{\frac{2K}{m}} $
8) $ x = a - \frac{ay}{b} $
9) $ m = \frac{y - c}{x} $
10) $ R = \frac{P}{I^2} $

Let me know if you'd like these written in a printable format or with extra explanation!