A dam wall
The Right Trapezoid
Cut a dam in half and look from the side. The water face is vertical, the back face slopes, the base is far wider than the top. That shape isn't a preference — it couldn't be otherwise. Why?
Three ways to see it
Water doesn't push on the wall with the same force at every depth. The deeper you go, the more water sits above you and the greater the pressure. That's why the wall thickens downward: every level is exactly as thick as the pressure at that level demands.
Switch layers — the scene stays put
What's really going on
The (a + b) / 2 in the trapezoid area formula isn't a shortcut — it is literally the width at the middle. If you are a wide at the top and b at the bottom and the change between them is even, then halfway down you are exactly the average of the two. That's why a trapezoid behaves like a rectangle of average width. In the dam this means: water pressure grows *evenly* with depth, so the required thickness grows evenly, so the section is neither a triangle nor a curve — it is exactly a trapezoid. The shape comes from the pressure being linear, not from architecture.
The equation
A = (a + b) / 2 · h
a and b are the parallel sides, h the perpendicular distance between them. Take the average of the two parallel sides — that's the width at the middle — then multiply by the height. The slanted side's length is absent, because it doesn't affect the area.
Same concept, elsewhere
Memorising one example gets you nowhere. You need to spot the concept wherever it turns up.
A retaining wall — soil, like water, pushes harder with depth, and the section is again a right trapezoid
The side panel of a staircase: one step high at one end, a full storey at the other
The cross-section of an irrigation channel — the vertical wall speeds the flow, the sloped face keeps the soil from sliding in
A cropped aircraft wing section: thick at the root, thinning toward the tip
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