Area of Parallelograms and Trapezia

You already know the area of a rectangle: \text{base} \times \text{height}, the number of unit squares that tile it. But the world is full of shapes that lean — a garden plot on a hillside, the sloping cross-section of a wheelchair ramp, a slice of a bridge support. How much surface do they cover?

Here is the beautiful trick: a leaning parallelogram secretly has the same area as a plain rectangle with the same base and height. Snip a triangle off one end, slide it to the other, and the lean vanishes — no area was added or lost. Once you see that, both the parallelogram formula and the trapezium formula fall straight out.

The parallelogram: base × height

A parallelogram looks like a pushed-over rectangle. Its area is the base times the perpendicular height — the straight-up distance between the two parallel sides, not the length of the slanted side:

A = b \times h

Why? Cut a right-angled triangle off one slanted end and slide it across to the other end. Nothing is added or removed, so the area is unchanged — but now the shape is an ordinary rectangle of width b and height h, whose area we already know is b \times h.

The trapezium: average the parallel sides

A trapezium has just one pair of parallel sides, of lengths a and b, separated by a perpendicular height h. Its area is the average of the two parallel sides, multiplied by the height:

A = \tfrac{1}{2}(a+b)\,h

Read it out loud: "half of a plus b, times the height" — the mean of the top and bottom edges gives the width of the equivalent rectangle. If a = b the two parallel sides are equal and the formula collapses back to the parallelogram's b \times h.

Both formulas measure the perpendicular height — never the slanted side:

See it

Step through the figure: cut the triangle off the parallelogram and slide it across to make a rectangle, then meet the trapezium with its two parallel sides.

Worked example 1 — mind the height

A parallelogram has a base of 10\text{ cm}. Its slanted side is 8\text{ cm}, and the perpendicular distance between the two parallel sides is 6\text{ cm}. Find its area.

The 8\text{ cm} slant is a decoy. Area uses the perpendicular height, which is 6\text{ cm}:

A = b \times h = 10 \times 6 = 60\text{ cm}^2.

Multiply by the 8 and you would get 80 — a classic wrong answer, because the slant leans across more than the true straight-up height.

Worked example 2 — a trapezium

A trapezium has parallel sides of a = 5\text{ m} and b = 11\text{ m}, held h = 4\text{ m} apart. Its area is the average of the parallel sides times the height:

A = \tfrac{1}{2}(5 + 11)\times 4 = \tfrac{1}{2}(16)\times 4 = 8 \times 4 = 32\text{ m}^2.

The average of 5 and 11 is 8 — so this trapezium covers exactly as much as a 8\text{ m} \times 4\text{ m} rectangle.

Worked example 3 — a real ramp

Seen from the side, a concrete access ramp is a trapezium: a short vertical face of 0.4\text{ m} at the top, a tall vertical face of 1.2\text{ m} at the bottom of the slope, and these two parallel edges are 3\text{ m} apart along the ground. How much cross-section area does the concrete fill?

A = \tfrac{1}{2}(0.4 + 1.2)\times 3 = \tfrac{1}{2}(1.6)\times 3 = 0.8 \times 3 = 2.4\text{ m}^2.

Multiply that by how long the ramp is (into the page) and you would get the volume of concrete to pour — which is exactly how builders estimate the job.

This is the single most common area mistake in the whole topic. For a parallelogram (and for a triangle) the h in A = b \times h is the perpendicular height — the straight-up gap between the parallel sides — and not the length of the sloping side.

The slant is always longer than the true height (it takes the long, leaning route across the gap), so using it makes your area come out too big. If a diagram gives you a slant length, treat it as bait: hunt for the little right-angle mark that shows the real perpendicular height, and use that.

The trapezium rule "average the parallel sides, then times the height" is doing something sneaky. Picture the graph of a straight, sloping line above the x-axis: the region under it between two points is a trapezium lying on its side — two vertical edges of different heights, a horizontal base between them. Its area is \tfrac{1}{2}(\text{left height} + \text{right height})\times \text{width} — the very same formula.

Chop any curvy shape into thin vertical strips, treat each strip's top as a little straight slope, add up all the tiny trapezia, and you have estimated the area under a curve. That idea has a name — the trapezoidal rule — and it is a first step toward integration. It is also exactly how surveyors measure the area of an oddly-shaped field: slice it into strips and add up trapezia. School geometry and calculus, quietly shaking hands.

Take any triangle and stand a second, upside-down copy of it right next to it. The two snap together into a parallelogram with the same base and height. So the triangle is exactly half of that parallelogram — which is why the triangle's area is \tfrac{1}{2} \times b \times h, half of b \times h. And notice: the trapezium formula \tfrac{1}{2}(a+b)h becomes the triangle's \tfrac{1}{2}bh the moment the top side a shrinks to 0 (the trapezium tips over into a triangle). All three formulas are secretly the same family — one idea wearing three hats.

See it explained