The family tree
The six shapes are not six separate things. They nest.
A square is a rectangle, a rhombus, a parallelogram and a kite all at once. A rectangle is a parallelogram with right angles. A rhombus is a parallelogram with equal sides. A parallelogram is a trapezoid with both pairs of sides parallel.
So a square legitimately answers to five names, and any formula for a more general shape works on a more specific one. A rhombus can be measured with the parallelogram formula or the diagonal one, and both give the same answer.
One warning worth stating: whether a parallelogram counts as a trapezoid depends on which definition a textbook uses. The inclusive one says yes, the exclusive one says exactly one pair of parallel sides. British and American conventions differ, and so do the words — a trapezoid in one is a trapezium in the other, meaning something else again.
The height trap
Base times height, for a parallelogram. The trap is which height.
It is the perpendicular distance between the two parallel sides — straight across, at a right angle. Not the length of the slanted side.
The slanted side is always longer, so using it always overstates the area. And it fails quietly: nothing about the arithmetic complains, the answer looks plausible, and it is simply wrong. A parallelogram with base 5, slant 4 and height 3 has area 15, not 20.
Push it further and the point becomes obvious. Hold the base and the slant fixed and lean the shape further and further over: the area shrinks towards zero while the side lengths do not change at all. Side lengths alone cannot determine a parallelogram’s area, which is exactly why the formula asks for something else.
This page refuses a height greater than the slant, because that combination describes no parallelogram at all.
Diagonals do the work
For a rhombus and a kite, the area is half the product of the diagonals — and it is the same formula for both.
The reason is that in both shapes the diagonals cross at right angles. That is the only property the formula needs, so it works for any quadrilateral with perpendicular diagonals whatever else is true of it.
The difference between the two is which diagonal gets bisected. In a rhombus both do, meeting at their shared midpoint. In a kite only one does.
That is also why a rhombus’s side comes straight from Pythagoras on the two half-diagonals, and a kite’s does not — there is no single pair of half-lengths to use.
The trapezoid formula
Average the two parallel sides, multiply by the height. ½(a + b)h.
Read as an average it stops being something to memorise: a trapezoid has exactly the area of a rectangle whose width is the mean of its two parallel sides.
That mean is also a real line in the figure — the midsegment, joining the midpoints of the two slanted sides. Its length is the average, which is a satisfying way to see the formula rather than derive it.
It also contains the other formulas. Set a = b and it becomes a parallelogram’s base times height. Set b = 0 and it becomes a triangle’s ½bh. Both are special cases of the same expression, which is a good reason to learn this one and not the others.
Where it gets used
Building and making. Flooring, roofing, fabric and land are rarely rectangular, and the trapezoid is the commonest correction.
Surveying. A plot with two parallel boundaries is a trapezoid, and the average formula is the standard method.
Numerical integration. The trapezoidal rule approximates an area under a curve by a row of trapezoids, and it works precisely because their combined area is easy to compute.
Engineering sections. Structural profiles are built from rectangles and trapezoids, and their areas feed into the second moment calculations that decide stiffness.
Sources and methodology
The formulas and classifications are curriculum standards; these are the references.
Method. Each shape asks only for the measurements its own formula needs, rather than offering a general form that quietly assumes something. The parallelogram refuses a perpendicular height larger than the slanted side and says why: the height is measured straight across, so it is always the shorter of the two, and accepting it would return an area that is too large with no indication anything was wrong. That engine is verified on every change against 114 hand-written assertions, including that a perpendicular height exceeding the slanted side is refused with a stated reason, and that a rhombus’s side is recovered correctly from its two diagonals. The count and the per-case breakdown are published on the formula verification page.
Read the guide
For a quadrilateral with no special properties, the Polygon Area Calculator works from the four corner coordinates.