DIY Fence Part 1: Math Formula For Board-on-Board or Louvered Fence

When building a privacy fence or gate with overlapping boards, most people simply estimate the overlap amount and number of boards needed. This often leads to uneven spacing or running out of material. In this video, Dr. David Zhang derives the exact mathematical formula for calculating the perfect overlap based on your gate width and board size.

The Formula

Given gate width W, board width B, and N boards, the overlap amount X is:

X = (NB - W) / (N - 1)

And the visible space between boards from one side: S = B - 2X

Original Contribution

Dr. Zhang has not found this formula presented or explained anywhere else. While the math is straightforward, having it explicitly stated saves builders significant trial-and-error time and material waste.

For more home improvement projects, check out how to do bathtub caulking and rearview mirror repair.

The Two Problems With Estimating Board Overlap

When building a privacy fence or fence gate, many people simply estimate the amount of overlap and number of boards. For a gate with seven boards, you first lay four boards in one layer using symmetrical spacing, then lay three boards on top of the first layer with symmetrical spacing.

The first problem is that it is easy to estimate roughly the amount of overlap when you have an odd number of boards. But when you have an even number of boards, it is not as easy because it is not symmetrical.

The second problem is that to estimate the board overlap amount, you need to place all boards first and then adjust each overlap amount. It would be nice if you could determine the exact overlap beforehand and nail one board at a time. This will save you time.

Setting Up the Algebra

I will derive the overlap formula using simple algebra for both odd and even numbers of boards. Assume the gate width is W, the board width is B, and the overlap amount is X.

The total number of boards is N, equal to 2M for an even number of boards — because both layers have the same number of boards — and 2M + 1 for an odd number of boards.

Deriving the Formula for Odd Numbers of Boards

When you have an odd number of boards, the first layer has M + 1 boards and the second layer has M boards, for a total of 2M + 1.

The spacing between each board in the first layer is B minus 2X. Width W satisfies this equation. Rearranging gives the overlap X for the odd-board case.

Deriving the Formula for Even Numbers of Boards

When you have an even number of boards, both layers have M boards. The spacing between each board in the first layer is still B minus 2X. Width W satisfies this equation. Rearranging gives the overlap X for the even-board case.

The Final Formula

Whether the total number of boards N equals 2M + 1 or 2M — odd or even — the result is one final unified formula. Once you know the exact amount of overlap, you may simply lay and nail one board at a time.

Original Contribution

I have not seen anyone else providing this formula. My videos always provide original and useful information. Share this with people you know who need it, and leave your own genius tips in the comment section below.