The fence post problem
A hundred foot fence at eight foot spacing has thirteen bays and fourteen posts. Divide and you get twelve and a half, round to thirteen, order thirteen posts, and the last bay has nothing to fasten to.
That off-by-one has a name in computing — the fence post problem, borrowed from exactly this — and it is one of four mistakes that account for most of what goes wrong in this category. The others are the pitch factor on a roof, the nominal size of timber, and the rule that every riser in a flight has to match.
Posts, bays and the extra one
Anything laid in a line has two counts, and they differ by one. The gaps between things — bays, spans, panels — are the length divided by the spacing. The things themselves — posts, uprights, ends — are always one more, because the run has two ends and both need something at them.
A hundred foot run at eight foot spacing: 12.5 bays, which rounds up to thirteen, and fourteen posts. The bays are not all eight feet, either. Thirteen equal bays across a hundred feet is 7 feet 8 inches each, which is fine, and rather better than twelve bays of eight and a short one of four at the end.
The same shape appears in joists, studs, balusters and rails. Rails multiply: three rails per bay across thirteen bays is thirty-nine rails, not thirty-nine feet of anything.
100 ft run / 8 ft spacing = 12.5 -> 13 bays
Posts = bays + 1 = 14
Rails = 13 bays x 3 = 39
Concrete for 14 holes = 16.1 cu ft (43 x 50lb bags)
Work it out: Fence Calculator →
A roof is bigger than the house under it
A forty by thirty house has a 1,200 square foot footprint. Put a 6/12 pitch on it and the roof is 1,342 square feet, because a sloping surface covering a given footprint is longer than the footprint by a factor that depends only on the pitch.
For 6/12 that factor is 1.118 — about twelve per cent. It climbs quickly: a 12/12 roof is forty-one per cent larger than the ground beneath it. Ignoring it on a steep roof is not a rounding error, it is most of a truckload of shingles.
Roofing is then sold in squares of a hundred square feet, three bundles to the square. With ten per cent waste the roof above comes to 14.76 squares and forty-five bundles. That covers the field only — ridge, hip and starter courses are separate, and they are the ones people forget when the field arithmetic was the hard part.
Footprint 40 x 30 = 1,200 sq ft
6/12 pitch factor = 1.118
Roof area = 1,342 sq ft
+ 10% waste = 1,476 sq ft = 14.76 squares
Bundles at 3 a square = 45
Work it out: Roofing Calculator →
A "2x6" is not two inches by six
Softwood is sold by its nominal size, which is what it measured before it was dried and planed. A two-by-six is actually one and a half by five and a half inches. A two-by-four is one and a half by three and a half. The nominal size is not a lie so much as a fossil, but it is a fossil that ruins a take-off.
It matters most where boards sit side by side. A twelve foot wide deck in "six inch" boards is not twenty-four boards, because the boards are 5.5 inches and there is a quarter inch gap between them. Twenty-six rows is the answer, and at sixteen feet each with waste that is 458 linear feet.
Board feet — the unit hardwood is sold in — use the nominal thickness and width but the actual length, which is its own trap. A board foot is 144 cubic inches: thickness times width times length, in inches, over 144. Ten eight-foot two-by-sixes is eighty board feet.
Work it out: Decking Calculator →
Stairs: every riser has to match
Stair arithmetic looks like division and is not. You cannot pick a riser height and divide the rise by it, because the answer will not be a whole number and the remainder has to go somewhere. Instead you divide, round to a whole number of risers, and then work the riser height back out from that.
A 108 inch total rise at a target of seven inches gives 15.4 risers. Round to fifteen and each riser becomes 7.2 inches — slightly taller than the target, and identical all the way up, which is the part that matters.
There is always one fewer tread than riser, because the top riser lands on the floor above rather than on a tread. Fifteen risers, fourteen treads, and at ten inches each a total run of 140 inches — nearly twelve feet of floor, which is usually the constraint nobody measured for.
An odd step out is the classic trip hazard, and it is why the riser height is a derived figure rather than one you get to choose. Stair rules also vary by jurisdiction, so check what governs the job before cutting stringers.
Total rise 108 in / 7 in target = 15.43 -> 15 risers
Riser height 108 / 15 = 7.2 in (all identical)
Treads = risers - 1 = 14
Total run 14 x 10 in = 140 in (11.67 ft)
Work it out: Stair Calculator →
Questions
How many posts do I need for a 100 foot fence?
Fourteen at eight foot spacing. A hundred divided by eight is 12.5, which rounds up to thirteen bays, and a run of thirteen bays has fourteen posts because both ends need one.
Why is my roof area bigger than my house?
Because the roof slopes. A 6/12 pitch makes the roof about twelve per cent larger than the footprint under it, and a 12/12 pitch makes it forty-one per cent larger. The factor depends only on the pitch.
What is a roofing square?
A hundred square feet of roof. Shingles are sold three bundles to the square, so a fifteen square roof is forty-five bundles of field shingles plus whatever the ridge, hips and starter course need.
How wide is a 2x6 really?
Five and a half inches, and one and a half thick. Nominal sizes describe the timber before it was dried and planed, and every softwood dimension is smaller than its name.
How do I work out stair risers?
Divide the total rise by the riser height you want, round to a whole number of risers, then divide the rise by that number to get the actual height. Every riser in the flight must be identical, which is why you cannot simply choose it.
How many treads for fifteen risers?
Fourteen. The top riser lands on the floor above rather than on a tread, so there is always one fewer tread than riser.