There are only two formulas to hold in your head, and they sort all three shapes. The whole game is picking the right one for what's in front of you.
The Trapezium: ½(a + B)h
Add the two parallel sides, multiply by the height, then halve. One — add the parallel sides (a + b). Two — multiply by the height h. Three — halve it. That's it.
A worked one: a trapezium with parallel sides 4 and 8, height 5. a + b = 12, then 12 × 5 = 60, then ½ × 60 = 30. Area = 30 square units.
The Rhombus and Kite: ½ × D₁ × D₂
These two share one formula because they share one feature: diagonals that cross at right angles. Multiply the two diagonals and halve. A kite with diagonals 6 and 10: 6 × 10 = 60, then ½ × 60 = 30. Area = 30 square units. A rhombus works the exact same way — it's just a kite with all four sides equal.
The Order Doesn't Matter, the Halving Does
You can halve first or last — ½ × 12 × 5 is the same as 12 × 5 ÷ 2. The one thing you can never skip is the halve. Forget it and your answer is exactly double the truth, which is the most common slip on these. Lock the ½ in.