The triangle is the only shape that will not wobble
Make a SQUARE out of four sticks pinned at the corners and push one side: it folds over into a lopsided DIAMOND without a single stick bending. Do the same to a TRIANGLE and nothing happens at all. Three side lengths fix the three angles, so there is nowhere for it to go.
That single fact is why roof timbers, cranes, electricity pylons, bridges and bicycle frames are full of triangles, and why a wobbling shelf is cured by screwing one diagonal across the back. A PYRAMID is the same trick standing up, which is part of why the Egyptian ones are still there.
Hexagons cover a floor for free
Only three regular shapes tile a flat surface with no gaps and no overlaps: the TRIANGLE, the SQUARE and the HEXAGON. A regular PENTAGON cannot manage it — try five-sided tiles on a kitchen floor and awkward slivers appear everywhere.
The hexagon is the pick of the three, because it walls off the most space for the least edge. Bees found that out long before anyone proved it, and so did cooling lava: the basalt columns of the Giant's Causeway in Northern Ireland cracked into rough hexagons all by themselves. Mathematicians only finished the formal proof in 1999.
Why a stop sign is an octagon
American road engineers in the 1920s agreed to sort signs by how many sides they had, with more sides meaning more danger. The CIRCLE was kept for railway crossings, the OCTAGON for stop, the TRIANGLE for give way and the DIAMOND for a warning.
The point was that a driver could read the sign from behind, or with snow plastered across the front, without seeing a single letter. Shape carries the message on its own. It is also a free counting lesson, since octa means eight, the same eight an octopus is carrying about.
The things that have to be round
A manhole cover is a CIRCLE because a circle is the one common shape that cannot fall down its own hole. Turn a square cover on its diagonal and it drops straight through onto whoever is below.
A soap bubble is a SPHERE because a sphere wraps the most air in the least skin, and the bubble is simply pulling itself as tight as it can. A drinks can is a CYLINDER for a related reason — curved walls hold pressure without buckling — with flat ends added so the things will stack. Even a traffic CONE is doing a job: all its weight sits low and wide, so a gust of wind cannot tip it.
Flat, solid, and the ones that are really pictures
The grid holds three different kinds of shape. There are flat ones you draw: the CIRCLE, SQUARE, RECTANGLE, OVAL, TRIANGLE, PENTAGON, HEXAGON and OCTAGON. There are solid ones you can pick up: the CUBE, SPHERE, CONE, CYLINDER, PYRAMID and PRISM.
Then there are the symbols. A STAR, a HEART, a CROSS, an ARROW and a CRESCENT are not really geometry at all — they are pictures everybody has agreed to draw the same way. A real star is a ball of burning gas and a real heart looks like a fist full of plumbing, which makes them a good thing to argue about while hunting for them.