trapezion

Greek trapeza meant a four-legged table or dining table — from tetra (four) and peza (foot). A trapezion was a small table. When Greek geometers named the quadrilateral with one pair of parallel sides, they chose the table shape — the flat top surface of a table viewed from above, with the legs spreading out unevenly. The name was observational and visual, not abstract.

The terminology becomes confusing in British vs. American usage: what Americans call a trapezoid (one pair of parallel sides) the British call a trapezium, and what Americans call a trapezium (no parallel sides) the British call a trapezoid. The confusion arose from different traditions of translating Euclid. Proclus, the 5th-century BCE commentator on Euclid, distinguished the two shapes, but the distinction was not consistently maintained across languages.

The trapezoid's area formula — ½(a + b)h, where a and b are the parallel sides and h is the height — is one of the fundamental geometric formulas. It generalizes both the rectangle (when a = b) and the triangle (when b = 0). Architecture uses trapezoidal shapes constantly: keystones in arches, trapezoidal fields on hillsides, and the frustum (truncated cone or pyramid, which has trapezoidal cross-sections) in construction.

Architectural keystone arches are trapezoidal: the central keystone is a trapezoid wider at the top than the bottom, allowing compressive forces to hold without mortar. Roman arches, medieval cathedral vaults, and modern stone bridges all rely on trapezoidal shapes. The table that gave Greek geometers their vocabulary is embedded in every arch.