Rhombus Perimeter

Four equal sides, so $P = 4s$ — arithmetically identical to a square's perimeter, because a rhombus is a square that has been pushed sideways. The distinction the two shapes draw is the one that catches people out everywhere in geometry: equilateral is not the same as equiangular. Equal sides do not force equal angles for any shape past a triangle. A rhombus has the sides of a square and not its corners; a rectangle has the corners and not the sides; a square is what you get when both hold.

Because leaning the shape leaves every side length untouched, the perimeter is fixed while the area is not. Hinge four 300 mm rods into a rhombus and the enclosed area sweeps from $s^2 = 900$ cm² when it is square down to nothing as it flattens, with $P = 1.2$ m the whole way. That is the isoperimetric idea in its smallest form — among all rhombi with a given perimeter, the square holds the most area, just as among all shapes whatever, the circle does. It is also why chain-link fence and expanding lattice trellis are built from rhombi: the pattern can rack and stretch without any wire changing length.

The working direction is usually $s = P/4$. Twelve metres of trim around a diamond-shaped panel means 3 m per side, and each mitre is 45° at the corner regardless of the lean, since the cuts follow the sides rather than the diagonals.

The trap on a real diamond-shaped object is that the side is not what you naturally measure. Reach for a tape on a diamond pane or a lattice opening and you will measure tip to tip — the diagonals, not the edges. The bridge between them comes from the four right triangles the diagonals make: $s = \tfrac{1}{2}\sqrt{d_1^2 + d_2^2}$. A pane 340 mm by 190 mm across the diagonals has sides of 195 mm and a perimeter of 779 mm, not the 1060 mm you would get by adding the diagonals themselves. The second thing to keep straight is that the perimeter tells you nothing about the area here — for that you need the diagonals or the lean angle, and the side length alone will never supply it.