Lightning Collection Area of a Rectangular Structure

A building is a bigger target than its roof. A leader that would have reached ground some distance away from the walls will instead attach to the structure, because the structure got close enough first. The equivalent collection area is the patch of ground over which that happens: the area which, if lightning fell anywhere on it, would have hit the building.

IEC 62305 gives it for a rectangular structure as the footprint, plus a strip three building-heights wide along each of the four walls, plus a quarter-circle of the same radius at each of the four corners. Add the four quarter-circles and they make one full circle, which is why the equation ends in $\pi(3H)^2$ rather than four of anything: $A_D = LW + 2(3H)(L+W) + \pi(3H)^2$.

Where the three comes from. It is the rolling sphere flattened onto level ground. For a structure of height $H$, the horizontal distance at which a descending leader is captured works out to about $3H$ under the standard's simplifying assumptions, and that ratio is what turns a three-dimensional attachment problem into a piece of plane geometry you can draw on a site plan.

The surprise in the arithmetic is how much bigger the answer is than the roof. A 50 m by 20 m warehouse 10 m to the ridge has a 1,000 m² footprint, 4,200 m² of side strips and 2,827 m² of corner circle — a total of 8,027 m², eight times the roof it sits under. And because the corner term grows as $H^2$ while the footprint does not grow at all, the ratio explodes for anything tall and narrow. A 30 m mast on a 2 m by 2 m base collects about $\pi(90)^2 \approx 25{,}400$ m² — six acres of ground, from a structure you could park a car on. That is the whole reason masts, chimneys, silos and church spires are assessed on their own rather than lumped in with the shed beside them.

Read the height equation backwards and the same fact appears as a square root. Solving $A_D$ for $H$ gives $H = [\sqrt{(L+W)^2 + \pi(A_D - LW)} - (L+W)]/(3\pi)$ — only the positive root is physical, the quadratic's other root being negative. Because the area grows as the square of height, the inverse is correspondingly insensitive: doubling the collection area asks for far less than a doubling of height.

What this form does not cover, and it is a real list. It is the simple rectangular case on effectively flat ground. It does not account for a hilltop or a slope — that is handled separately, and crudely, by the location factor $C_D$ in the strike-frequency calculation. It does not account for nearby taller structures, which genuinely shield, and again $C_D$ is the only place that enters. Most importantly, for a stepped, L-shaped or complex roof the standard's graphical method — actually drawing the $3H$ envelope around the real outline, using the local height at each part — will give a different and better answer than plugging a bounding box into this equation, sometimes substantially different. A building with a 40 m tower on one corner of a 6 m warehouse is not a box, and treating it as a 40 m box overstates the area badly while treating it as a 6 m box understates it.

The area is not an answer on its own. It exists to be multiplied by a ground flash density and a location factor, which is the next page, and even that is only the first line of a risk assessment.