When measuring many things with a view to costing them, such as any builder (client, surveyor included) might well do, it is surprisngly common that one needs to measure an area AND to measure the perimeter of the same surface.

For example, if replacing a worktop in a kitchen, there is the bulk work of preparing the surface and then the 'fiddly bit' along the back edge and possible a side. Indeed, there may be different edge effects. In quantity surveying, these actions are often labelled as being 'extra over'; that is, you've measured for the worktop area and proced that as a liece of work, but then the extra work at the edges follows.

In brickwork this applies at all openings (door and windows) and in window fitting it is quite likely that the window itself is actually priced in terms of area and perimeter.

Funnily enough, if you assume that all angles in a complicted 2-d shape are right angles, then the perimeter is likley to be the same as for a simple enclosing rectangle. This then would ignore the extra procing on a very odd shape if there were to be more than four corners—and a decent extimator would want the opportunity to add cost for the implied additional work—but in general we might well assume that anything with dimensions a and b had area ab and perimeter** 2(a+b)**, priced differently. let's say that the area is priced at A and the perimeter additional costs at B. Then the cost formula for the work will be abA + 2(a+b)B.

This general rule applies to extensive and expensive work such as will be contemplated by anyone looking to insulate their home. we might develop a matcing formula for three dimensions but in practice the floors and ceilings require different activities, so this is not useful. Indeed, if we consider insulating a house on the outside, then we must consider the required area as the external wall area, plus edge effects at the top and bottom and perhaps at corners, less the space made by the openings, basically, windows and doors, but for each of these we have to add 'extra over for the added perimeter. There are other difficulties like overflow pipes, downpipes, external cabling, all of which have a linear measurable element).

So an estimate for insulating the area of a house might well start with a total hross wall area (house height to eaves multiplied by total perimeter), then measure the corners (which may be more than four, if there are things like bay windows, in which case there may be different work attached to external corners and internal corners); i say 'measure' rather than count becuse you cannot assume that all corners are the full house height. where the additional external insulation meets the ground there will be a linear detail and there will be another, probably different, one where the additional work meets the eaves. It might be that thsi includes removing and refixing the gutters. For each door and window (building openings) there is a subtraction for the saved area but an extensive addition for whatever extra work is required around the edges, which might include work similar to that of the gutter, moving the window or door. We cannot assume that horizontal and vertical edges are treated the same.

Then for all those extra (linear) things like pipes and cables we have a poiint effect (an overflow pipe, an aerial) and a linear effect, (pipes and cables) - and again. there needs to be an understanding what the design needs are and therefore what the cost consequences are.

All of this applies to any surface covering, even somethign relatively simple like paintwork.

There is a related problen to do with painting general structures of apparent area and actual area.

Q1. Consider painting a rectangular window dimensions a and b . That's a across and b vertical. The window is to be fitted landscape (so a>b) The framing is, say, 75mm top and bottom and 50mm at the sides. There is an additional bevelled edge around the glass at 45º, 10x10mm. Produce an estimate of the area and perimeter elements, in metres and square metres.

Q2. Add an extra for the window being divided into four smaller panes with a ('beading')detail roughly 10x10mm and estimate the perimeter of that cross-section as 28mm.

Q3. Now divide the window into 12 panes not 4 and work out the extra over component t consider.

You might think that thsi is trivial. I had an equivalent task for a 'tank farm' on an oil refinery. The tank farm was approaching a square kilometre of large cylindrical tanks of different sizes (and contents). Each tank had external structure (it's the inside that needs to be smooth) amounting to, I eventually realised, about twice as much more area thatn the cylinders implied. I had to approve (agree or disagree with) the sub-contractor's figure for the area so as to agree or disagree with the money he wanted paying. The contract for the refinery meant that I had to do this for £1-300,000 per day, every day just to achieve my share of the work, so I reckoned on this occasion I could justify a half day getting the number right. So I took two examples, a short squat cylinder andf a long thin one (basically a chimney) and worked out fine detail for a section of each of these, which gave me a multiplier of between 3 and π to apply to what I thought of as the assumed perimeter of any tank. It was a little less for the squat one and a bit more for the chimney, mostly because of the ladders (that I climbed to check the painting had been done; the chimney waved in the wind). I ended up thinkiing that the subcontractor had painted a few other things not in the contract with the same paint and produced a pile of calculations that showed he was at least 15% over the top. The working rule was that if my figure agreed with the offered figurre by within 10%, that was okay, otherwise we took my figure (almost always smaller, but not necessarily; if my figure was bigger we physically went to check that the job had been completed). And the sub-contractor got told off by me (typically, I'd be a generation younger and just as angry). If he wanted to extend the argument, we'd go look at the work (but I usually couldn't afford the time, such was the pace of the job) and i'd point out was measured and what was not. This very often showed that the contractor hadn't looked at the drawings, hadn't read his contract and (not or) didn't really know what it was he'd been supposed to be doing. Add into that that the contract languiage was English but the 500 subcontractors were Portuguese and you can see that, quite soon, our office had a reputation for being difficult.

And usually right. the job was back in 1978, so you could probably add another zero to the contract in current terms. Did you that this sort of arithmetic would be boring?

Q4. Describe how you might measure the volume of a piece of tarmac road, bearing in mind that the tarmac-laying process is very much more expensive than that of the sub-base, so the thickness actually laid matters a lot. There is a rough guide given by the volume of tarmac delivered, but that is only an upper bound, since the finished volume has been compressed. There are surveyors marks at ten metre intervals indicating where the finished surface is supposed to be. Usually these indicate a metre above the target height, so they serve as sight lines. For example, the digger driver working on an embankment will have a sight line showing him the expected bank angle, while down at the roadside (when the embankment is finished) there will be another set of guideposts showing the finished road height.

DJS 20220730

A1. Area to paint = 2a (0.075) + 2b(0.05) + 2((a-0.1)+(b-0.15))(0.014) = 0.178a + 0.128b -0.042

which, you notice, inculdes no ab terms.

External perimeter is 2(a+b) and internal perimeter is 2((a-0.1)+b(0.15)) so total perimeter is 4(a+b) - 0.5

A2. You have two lengths of bead, a-0.1 and b-0.15, with some small area saved at the crossover (which I shall ignore at first), so the area to be painted is about 0.028(a+b-0.25). if youconsider the intersection, you might subtract 10mm from the working length, so the additional area to paint is 0.028 (a+b-0.26), a very small change. But you could very well argue that the extra over for this work is not an area but a length.

A3. So this requires two beads of one length, a-0.1 and three of the other b-0.15 (or the other way around if the window was portrait). There will be 6 crossovers, so the area is 0.028 multiplied by the total length, 3(a-0.1) + 2(b-0.15) - 6(0.01)

A4. There are two processes, before and after the tarmac laying. Both require there to be a survey of a cross-section of road. A horizontal string is set across the road at a known height (there are usually surveyor's boundary sticks that everyone treats as valuable, every 10 metres). Each crosssection is measured (with a stick, downwards from the string). The finished level is likely to be much easier to measure and will be within the design tolerances (surprisingly tight), so it is the bottom surface that is iimportant to get right – and you can go and remeasure the finished surface afterwards. So the measurements, however done, have to be agreed between the contract parties.

What you do with the measurements by hand is a 3-d version of numerical approximations for an integration (Simpson's rule, but look at Newton-Cotes too) , amounting to giving the intermediate measurements far more weight than the perimeter ones. I remember producing a formula with the usual 1s and4s but then having many measurements multiplied by 6 or 9 becasue of the interpolation in the other direction. Then I realised that every intermediate measure should be done at least twice, even that we perhaps ought to be measuring twice as much (5m intervals along the road, 50cm across each section). What looked at first as regualr planes ('flat' surfaces) turned out to be a long way from flat enough to assume anything very much. When you cost the time of the culculation against the value of the work being measured, you realise that your time is pretty much irrelevant until you can show that your result is precise enough to defeat argument.

The designed volume and the actual volume are very different but the various contractors are paid for work as finished. There can be circumstances where the client agrees to pay only what is designed, just as there can be circumstances where a payment is for the completed volume. So, while the material supplier (of tarmac) would be paid on tonnage delivered to site, the contractor who puts the tarmac down — in the right place to the right depth and consistency with a finish suitable for coping with the intended traffic and no bumps — has quite a different thing he wants to be paid for; his work, done right has significant value. Done wrong and he has to, often, take it up and try again. Pricing for those eventualities is a whole different game.

These days, you'd use a spreadsheet or a purpose-built surveying tool. what you'd not know is whether the numbers were in any sense correct.