Quadrilaterals: trapezoid, parallelogram, rhombus

A rectangle has an area equal to the product of its sides, because its sides are perpendicular. A trapezoid, a parallelogram and a rhombus do not enjoy that: their sides lean, so multiplying two lengths gives too much. All three formulas in this lesson come back to the same trick: cut the figure so that a rectangle or two triangles fall out of it.

Who is who among quadrilaterals

The names form a ladder, not a list of separate boxes:

• A trapezoid has at least one pair of parallel sides. Those two are its bases, the other two its legs.
• A parallelogram has both pairs parallel, so it is a special trapezoid.
• A rhombus is a parallelogram with all sides equal.
• A rectangle is a parallelogram with right angles, and a square is both a rhombus and a rectangle.

So every square is a rhombus but not every rhombus is a square — and that is why the parallelogram's area formula also works for a rectangle.

The area of a trapezoid

The height $h$ is the segment perpendicular to both bases. It is not the leg, even when the drawing makes the two look alike.

Add a copy of the trapezoid rotated by $180^\circ$. What appears is a parallelogram of base $a + b$ and the same height $h$, and our trapezoid is exactly half of it:

So the bracket holds the sum of the bases, not their difference — and the division by two is there because we counted two trapezoids at once.

Read backwards, the same formula answers the question about a missing base. Since $2P = (a + b)h$,

The area of a parallelogram

Cut the triangle off one leg of a parallelogram and attach it on the other side. What appears is a rectangle of base $a$ and height $h$ — with exactly the same area:

This is where the mistake happens: the height is not a side. In the drawing above the area is $7 \cdot 4 = 28 \ \text{cm}^2$, not $7 \cdot 5 = 35 \ \text{cm}^2$. The perimeter, on the other hand, is computed from the sides alone: $2 \cdot (7 + 5) = 24 \ \text{cm}$.

The rhombus and its diagonals

A rhombus has all sides equal, so its perimeter is simply $4a$. Its area can be found as in any parallelogram, but usually it is the diagonals that are known. In a rhombus they are perpendicular and bisect each other, so the rhombus falls into four congruent right triangles with legs $\tfrac{e}{2}$ and $\tfrac{f}{2}$:

The $3$, $4$, $5$ triple is no accident here: the side of a rhombus is the hypotenuse of a triangle built from half-diagonals, so the Pythagorean theorem ties the side to both diagonals.

Angles and diagonals

Any diagonal cuts a quadrilateral into two triangles, and a triangle has an angle sum of $180^\circ$. Hence at once:

In a trapezoid the angles on the same leg sit at two parallel lines cut by a third, so they add up to a straight angle:

In a parallelogram that holds for every pair of adjacent angles, and opposite angles are equal. Knowing one angle of a parallelogram therefore means knowing all four.

Common mistakes

• Multiplying two sides of a parallelogram — the area uses the height, which is shorter than the slanted side.
• Subtracting the bases of a trapezoid instead of adding them — the formula holds $a + b$, because two trapezoids form a parallelogram with exactly that base.
• Forgetting to halve the product of a rhombus's diagonals — $e \cdot f$ is the area of the rectangle around the rhombus, which is twice too much.
• Confusing the leg with the height — the height is always perpendicular to the base and usually lies inside the figure.
• Assuming the diagonals of a parallelogram are equal — they are equal only in a rectangle; in a rhombus they are perpendicular instead.
• Computing a rhombus's perimeter from its diagonals — the side has to be found from the Pythagorean theorem first.