Area — triangle, parallelogram, trapezium, composite shapes: Visual walkthrough

We only ever assume this one seed:

Everything below is a game of cut-and-slide that turns a shape into this one seed. See 1.2.1-Basic-shapes if the words "rectangle" or "right angle" feel shaky.

Step 1 — Meet the perpendicular height (the thing that actually matters)

WHAT. Before any cutting, we fix the one measurement every formula depends on: the perpendicular height. Take a slanted side and a base line. The height is not the length of the slant — it is the straight up-and-down gap between the base and the top, measured along a line that meets the base at a right angle.

WHY. Area counts how many unit squares stack up. A leaning shape does not cover more floor just because its edge is long; only the vertical squeeze stacks squares. So the honest measurement is always the perpendicular gap $h$ , never the slant $s$ .

PICTURE. The cyan slant $s$ leans over. The amber $h$ drops straight down and hits the base at the little right-angle square. They are different lengths — that difference is the whole story.

Figure — Area — triangle, parallelogram, trapezium, composite shapes

Step 2 — The parallelogram loses a triangle, gains a rectangle

WHAT. Take a parallelogram: base $b$ along the bottom, perpendicular height $h$ , leaning to the right. Slice straight down from the top-left corner to the base, creating a right triangle on the left. Cut it off and slide it all the way to the right.

WHY. The leaning left edge and the leaning right edge are equal and parallel (that is what "parallelogram" means). So the triangle we cut from the left fits exactly into the notch on the right — nothing overlaps, nothing sticks out. The area cannot change: we only moved a piece.

PICTURE. Left frame — the amber triangle before the move. Right frame — after sliding, the outline is now a clean rectangle of width $b$ and height $h$ .

Step 3 — The triangle is half of a parallelogram

WHAT. Take any triangle with base $b$ and perpendicular height $h$ . Make an identical copy, rotate the copy $18 0^{\circ}$ , and glue it to a side of the original.

WHY. Two identical triangles, one flipped, always lock together into a parallelogram — the rotated copy's slant edge matches the original's slant edge perfectly. The joined shape has the same base $b$ and the same height $h$ , because flipping never changes how tall or wide a shape is.

PICTURE. The cyan original triangle plus its amber flipped twin snap into a full parallelogram, base $b$ , height $h$ .

Step 4 — Any base works (and where the height goes when the triangle is obtuse)

WHAT. You may pick any of the three sides as the base. The formula still gives the same area because area is a property of the shape, not of your choice.

WHY. For a right or acute triangle the height lands inside the base. For an obtuse triangle (one angle bigger than $9 0^{\circ}$ ), the perpendicular from the far vertex lands outside the base — you must extend the base line to catch it. The number $h$ is still the straight perpendicular gap; it just meets an extended line.

PICTURE. Left — acute triangle, height foot inside. Right — obtuse triangle, the base extended by a dashed cyan line so the amber height can reach it.

Step 5 — The trapezium: two copies make a parallelogram

WHAT. A trapezium has two parallel sides of lengths $a$ (top) and $b$ (bottom), separated by perpendicular height $h$ . Copy it, rotate the copy $18 0^{\circ}$ , and attach it along a slanted side.

WHY. After the flip, the short side $a$ sits beside the long side $b$ , so the combined bottom edge of the new shape has length $a + b$ . The rotated copy's parallel sides stay parallel, so the union is a parallelogram of base $(a + b)$ and the same height $h$ . Two trapeziums built it, so one trapezium is half.

PICTURE. Cyan original trapezium + amber flipped copy → a parallelogram whose base is the full $a + b$ and whose height is $h$ .

Step 6 — Degenerate cases: watch the formulas collapse gracefully

WHAT. Push each shape to a limit and check the formula still tells the truth.

WHY. A formula you trust must survive its edges. If it gave nonsense when $a = b$ or $h \to 0$ , we would not trust it in the middle either.

PICTURE. Three shrinking cases side by side: trapezium with $a = b$ becoming a rectangle; trapezium with $a = 0$ becoming a triangle; any shape with $h = 0$ becoming a flat line of zero area.

$a = b$ : $\frac{1}{2} (a + a) h = ah$ — the trapezium is a rectangle. ✓
$a = 0$ (top side shrinks to a point): $\frac{1}{2} (0 + b) h = \frac{1}{2} bh$ — the trapezium is a triangle. ✓ The trapezium formula secretly contains the triangle formula.
$h = 0$ : every formula gives $0$ — a flattened shape covers no floor. ✓

The one-picture summary

One family, one seed. Rectangle → parallelogram (slide a triangle) → triangle (halve a parallelogram) → trapezium (double then halve). Every arrow is a cut-and-slide that keeps area honest, and every result is measured with the same perpendicular height $h$ .

Recall Feynman retelling — say it back in plain words

I only ever trusted one thing: a rectangle's area is its two sides multiplied, because that many little squares tile it. A parallelogram is a rectangle that leaned over — I chop the sticking-out triangle off one end and slide it into the gap on the other end, and out pops a rectangle of the same base $b$ and the same straight-up height $h$ , so its area is $b h$ . A triangle is just half of that: two copies of it, one flipped, snap into a parallelogram, so one triangle is $\frac{1}{2} bh$ . A trapezium with a short top $a$ and long bottom $b$ — I copy it, flip it, and the two lock into a parallelogram whose base is $a + b$ ; one of the two is $\frac{1}{2} (a + b) h$ , which is the same as saying "use the average width $\frac{a + b}{2}$ ." The height is never the slant — it is the straight perpendicular gap, and if I only know the slant $s$ and its angle I take $h = s sin θ$ . Every formula collapses correctly at the edges: equal parallel sides give a rectangle, a zero top gives a triangle, and zero height gives zero area. Same seed, four shapes.

Recall Quick self-test

Why is parallelogram area $bh$ and not base times slant? ::: The cut-and-slide turns it into a rectangle of height $h$ (the perpendicular gap); the slant is longer and would overcount. How does the trapezium formula contain the triangle formula? ::: Set $a = 0$ : $\frac{1}{2} (0 + b) h = \frac{1}{2} bh$ . A slant side of $7$ leans at $6 0^{\circ}$ . True height? ::: $7 sin 6 0^{\circ} = 7 \cdot \frac{3}{2} \approx 6.06$ .

Related paths: this "shape becomes a rectangle" trick is the seed of integration as area; for the round cousin see 1.2.13-Circle-area-and-circumference; outlines and edge-lengths live in 1.2.11-Perimeter.