The area of a trapezoid

A quadrilateral with one pair of parallel opposite sides is a trapezoid¹. We find the area of the one with vertices $A(0, 0)$, $B(6, 0)$, $C(5, 4)$ and $D(1, 4)$.

The formula

The side $AB$ lies on $y = 0$ and the side $DC$ on $y = 4$, both parallel to the $x$-axis. Their lengths are $AB = 6$ and $DC = 4$, and the gap between the two lines is the height $4$.

The formula becomes clear once a second copy of the trapezoid is turned upside down and set beside the first. Together they make a parallelogram whose base is the sum of the two parallel sides, of area $(a + b)h$, and the trapezoid is exactly half of it.

Seen through the midline

The segment joining the midpoints of the two non-parallel sides is the midline.

Its length is $5$, which agrees with $\dfrac{a + b}{2} = \dfrac{6 + 4}{2} = 5$. Writing $m$ for the midline gives $S = mh = 5 \times 4 = 20$: the trapezoid has been replaced by a rectangle as wide as its midline.

The shoelace formula gives the same

Tracing $A \to B \to C \to D$ gives $0 + 24 + 16 + 0 = 40$, and half of that is $20$.

The ratio in which the diagonals cut each other

The diagonals $AC$ and $BD$ meet at $(3, 2.4)$, and that point divides $AC$ from $A$ in the ratio $AB : DC = 3 : 2$. The ratio of the two parallel sides passes straight over to the ratio in which the diagonals are divided.

Triangles of equal area

The triangles $\triangle ABD$ and $\triangle ABC$ share the base $AB$ and the height $4$, so their areas are equal, both $\dfrac{1}{2} \times 6 \times 4 = 12$. Between two parallel lines the area of a triangle does not change wherever the apex is placed.

Linking the triangle and the parallelogram

The formula for a trapezoid is shaped so as to join those two together.

The four lines on the graph are the four sides, the line $y = 2$ contains the midline, and the large dots are the four vertices together with the ends of the midline.

1. Trapezoid, Wikipedia