\( P=2 \pi r+2 L+2 r \quad \) for \( L \)

To solve for \( L \) in the equation \( P = 2 \pi r + 2 L + 2 r \), first, you can isolate \( 2L \) by subtracting \( 2 \pi r \) and \( 2r \) from both sides, which gives you \( 2L = P - 2 \pi r - 2r \). Now, simply divide everything by 2 to find \( L \): \[ L = \frac{P - 2 \pi r - 2r}{2}. \] This equation reveals how the perimeter \( P \) relates to both the radius \( r \) and the length \( L \). When dealing with equations like this, a common mistake is forgetting to keep track of the terms when moving them around. Always ensure you perform the same operation to both sides of the equation and check your math after each step. A helpful tip is to rewrite the equation in a simpler form before starting your calculations; this can often clarify your next steps and help avoid errors.