Introduction: Getting Things Even
It has come to my attention that many people have difficulty figuring out how to space objects evenly other than by guess or estimate. That can lead to problems in some cases. Most people who make things or just do thing for themselves will occasionally want to evenly space objects along or within a certain distance. It may be spacing holes along a board, supports under a shelf, fence posts along a border, pickets on a fence, flower pots on a porch, or spacing trees in an orchard. Here's a mathematical way to get the spacing just right.
It helps me to draw a diagram and label the dimensions. If it helps you, do it.
Step 1: Case 1:
CASE #1: If you want to place objects (even if they're holes) at both ends of a space and have others evenly spread between them, subtract the width (or diameter) of the objects (actually you're subtracting half the width twice, but it's the same thing), like a fence post or flower pot, from the total length. This gives you the center-to-center distance of the end objects. Then divide by the number of spaces you want (one less than the total number of objects). Alternately, you can place the two end objects wherever you want them and measure from the center of one to the center of the other. (You will get the same result if you measure from the right side of one to the right side of the other - or the left sides of both.) You can use this method to figure the spacing of hooks to hang baskets from your porch roof, install lights along your sidewalk or space plants in your garden.
As an example, if you wanted to put a number of 12 inch flower pots along a 15 foot porch (180 inches long), you would subtract 12" from 180", leaving 168". This is actually the distance from the center of a pot on one end to the center of a pot on the other end. Then divide by the number of spaces (one less than the total number of pots including the ones at each end, or one more than the number of objects between those on the ends). Let's say that you wanted a total (including the end pots) of 4 pots. That means there would be 3 spaces so divide 168" by 3 and get 56", which is the distance from the center of each pot to the center of the next one. Subtract the width of the pots (12") to get the spacing between the pots (44"). If you add up all the dimensions you will get the exact length you started from. In this case --- 12" + 44" + 12" + 44" + 12" + 44" + 12" = 180" (If you don't, you did something wrong.) If you wanted to add another pot and still keep equal spacing, divide the 168" by 4 to get 42", then subtract the size of each flower pot (12") to get a spacing of 30" between pots.
A similar method would be used to figure out the spacing of fence posts along a property line, where there has to be a post at each end.
Step 2: Another Example of Case 1:
Another example of Case 1 is if you want to have a certain number of evenly-spaced holes in a piece of wood, metal or anything else, mark the centers of your two end holes and measure that distance. Add the size of each hole to that and divide the total by one less than the total number of holes you want (one more than the number of holes between the end holes). This will give you the center-to-center distance for perfect spacing. You can either measure out and mark each space or use a set of dividers or a compass (the kind used to make circles and arcs, not the kind for finding magnetic North) to mark your project.
Step 3: Case 2:
If you want to evenly space objects between two existing points, measure the space you have to fill and the size of the objects you want to place in that space. Add the two together and divide by one more than the number of things you want to place between them.
As an example, say you want to put fence posts between two existing pillars. The distance between the pillars is 18'-4" (which is 220 inches) and you're using 4x4 posts (which are actually 3-1/2" x 3-1/2"). Let's also say that you don't want the posts any more than 8 feet (96") apart.
Add the width of your object to the total space you want to fill. In the example given, it would be 223-1/2". This is distance "A". Regardless of the actual size of the end posts, columns or buildings, this will give you equal spacing for posts to be added between them. See the drawing.
Add the width of your maximum spacing to the width of your object. In the example, it would be 96" + 3-1/2" or 99-1/2". This is distance "B"
Divide the distance "A" by the distance "B" to find the number of spaces you will need. Obviously, this is more important when longer distances and more spaces are involved. In the example, this would be 223.5 / 99.5 = 2.24623. Since this is more than 2 and you want equal spaces between the objects, it means that you will need more than 2 spaces (which means you'll need three, since a quarter or a half a space wouldn't be equal to the whole ones). This tells you how many spaces there need to be between objects you need to fill the distance without going over the maximum spacing you set. Since there is a space between each object (posts) and the end, you'll need one less object than spaces (in this case, two posts). If you already know how many objects you want, you can ignore this step. (Notice that if you just put in one post between the two existing posts, there would be two spaces 108.25" between them --- 220" = 108.25+3.5+108.25. This is more than the 8 foot maximum we wanted.)
Divide distance "A" by the number of spaces between objects. In our example, this is 223.5" / 3 = 74.5". This is the center-to-center distance of the posts (or flower pots on your porch, holes in your board, or pickets on your fence). The spacing between the objects is that distance less the width of your objects. From the example, 74.5" - 3.5" = 71". This is less than the 8' maximum we wanted between posts and, if you add them all up, you'll see that it comes out exactly right; 71" +3.5" + 71" + 3.5" + 71" = 220" -- the exact distance we wanted to fill between the posts.
Step 4: Another Example of Case 2
This is also how to calculate how many pickets we need and their spacing between each post. The space from above is 71" and the pickets we intend to use measure 4-1/2" wide, so add the 4-1/2" to the 71" to get distance "A", then divide by 4-1/2 (4.5). This gives a result of 16.7777, which means that 17 spaces won't fit. Without cutting one or more of the pickets down, The most spaces which include the pickets themselves, that you can have (including the one past the post that you added) is 16. The actual number of pickets will be one less, since they will go between the posts, not covering one of them.
Now, if you divide the 75-1/2" by 16, you'll get 4.71875" (which is more than the 4.5" width of the pickets, so they will fit) then subtract the width of the pickets to get the spacing between them (4.71875 - 4.5 = 0.21875). Unless you're planning to use extremely accurate measuring devices, you're probably going to want to convert this to 16ths or 8ths, so just multiply this by 16 (or 8) to find out how many sixteenths (or eighths) are in 0.21875 (or whatever decimal you got from your measurement), which is 3.5 - 16ths. At this point, you could go to either 3/16" or 4/16" (1/4") spacing between your pickets. If you use 1/4" spacing, the last picket won't quite fit (there won't be any spacing between it and the previous picket). If you use 3/16" between them, your last picket will theoretically have 11/16" too much space between it and the post. You have to decide how you want to handle this, depending on your equipment, how fussy you are going to be, and how good your pickets are. If you use 3/16" spacing, or even 1/8" between them, and work towards the center from each post then adjust the space between the pickets for the last 5 or 6 pickets, no one will ever notice the difference... not even you.
Step 5: Watch Out!
In reality the pickets won't be exactly 4-1/2" wide - or even straight! Wood, as purchased from a lumber yard, simply isn't all that uniform. From experience, unless you have ripped your pickets to a specific width, you're going to find at least a 1/8" variation between the narrowest and widest ones. And it will probably be more. I've also purchased many pickets that had a 1/2" or more crook or kink (side-to-side warping) in them. If you want to find out the real (average) width of your pickets, lay them out, side by side, and measure the total width without any spacing between them and divide by the number of pickets. Use that number for your calculations. If you don't, by the time you work with more than a few of them, any errors will add up and you'll find that things don't work out the way you planned.
If you want the pickets to be further apart, decrease the number of pickets used, because you want fewer, wider spaces. For example, if you used 14 pickets between posts, the center-center distance would be 75.5 / 15 = 5.03" and the spacing between pickets would be 0.5333 or about 1/2". Always remember to divide by one more than the actual number of pickets you intend to use. If you only want to use 13 pickets, each picket would be spaced 75.5 / 14 = 5.39286" from the beginning of the previous one and you would need 0.80929 or about 3/4" between pickets. Dropping to 12 pickets would increase the space between them to just over 1-1/4" (1.3076"). Just remember to measure the ACTUAL average width of your pickets the way I described above.