A regular octagonal tabletop has a side of 15 cm, and the perpendicular from its centre to the middle of any side measures 18.1 cm. What is the area of the tabletop, in square cm?

Join the centre to every corner and the octagon falls into 8 equal triangles, each with base 15 cm and height 18.1 cm. That gives area = $8\times (1/2)\times 15\times 18.1=1086$ sq cm, which is the standard result (1/2) $x$ perimeter $x$ apothem = $(1/2)\times 120\times 18.1$. The trap $2172=120\times 18.1$ drops the 1/2; $135.75=(1/2)\times 15\times 18.1$ gives a single triangle and never multiplies by 8; 120 is the octagon's perimeter, a length quoted where an area was asked.