The Outer Tenth of a Disc Is a Fifth of Its Area

Doubling the wall does not double the metal

Taking the same 60 mm tube from a 3 mm wall to a 6 mm one lifts the steel cross-section from 537 to 1,018 square millimetres — a factor of 1.90, not 2. The reason is that the second three millimetres sit further in, on a smaller circumference, so each one contributes less area than the one outside it. Anyone estimating the weight of tube stock by scaling the wall thickness is therefore over-ordering slightly, and anyone estimating a thin-wall tube by the same shortcut is close enough that the error never shows. The thicker the wall relative to the diameter, the worse the shortcut gets.

Same shape, opposite conclusion for flow

The bore is the same square law read the other way, and it is why a modest increase in pipe diameter buys so much capacity. Going from a 27 mm bore to a 30 mm one — eleven per cent wider — raises the cross-section by twenty-three per cent, and the flow at a given pressure drop rises faster still, because the fluid near the centre moves faster than the fluid near the wall. That is the argument for oversizing a run of pipe rather than adding a pump, and it also explains why a build-up of scale that narrows a pipe by a couple of millimetres costs far more flow than the thickness suggests.

Frequently asked questions

Is a running track lane the same problem?

Related, but the quantity that matters there is the circumference rather than the area. Each lane outward adds two π times the lane width to the lap distance, and that increment is the same whatever the radius — which is why the staggered start lines on a track are evenly spaced and why the stagger does not grow for the outer lanes. Area is what you need for surfacing material and paint; circumference is what you need for fairness. Confusing the two is the fastest way to order the right amount of tarmac and the wrong starting positions.

How do I get the area if I only have diameters?

Halve them first and keep halving them first. Pipes, washers and tubes are almost always specified by diameter, and dropping the halving step multiplies the answer by four rather than by two — a mistake large enough to be obvious in a total and small enough to survive a quick glance at one figure. A tube described as 60 by 3 usually means a 60 mm outside diameter with a 3 mm wall, so the two radii are 30 and 27, not 60 and 57. When a supplier's table gives a mass per metre, that is the fastest cross-check available: work out the annulus, multiply by the density, and see whether the two agree.