As Adam and Jamie are constructing the cube-shaped lead balloon, the narrator (Robert Lee) states that the second square of foil is cut into equilateral triangles. It's actually cut into 4 isosceles triangles, with angles of 45º-45º-90º. He probably meant to say "equal-sized" triangles, which is correct.
When making the smaller aluminum foil balloon, in the shape of a platonic solid, Adam refers to the overall shape as a polygon. A polygon is a flat 2D shape such as a square or triangle, a 3D solid shape such as the one Adam was making is called a polyhedron, not a polygon.