# Constructing a trapezoid using the side lengths

Lloyd left a comment on a post of mine yesterday, asking:

how do you draw a Irregular quadrilateral trapezoid with fixed dimensions for the two parallel bases and the two legs with no angles given using geometry tools?
top base= 328
bottom base= 223
left leg =220
right leg= 215

How would you answer this question? It’s not trivial. You’ll quickly find that if you do a straight-edge and compass construction, you’ll need the height of the trapezoid.

If we let a, b, c, and d be the side lengths of a trapezoid with a and c as the bases, can we express the height h as

$h=\sqrt{b^2-\left(\frac{d^2-b^2-(c-a)^2}{2(c-a)}\right)^2}$

This number is constructable, but would take some work to actually construct it on paper. Perhaps we can return to that particular question later. For now, we can let GeoGebra show us the general idea. I’ve made this applet in which you can change the side lengths and the trapezoid will be constructed. I used the height formula above to calculate the height, and the applet shows this value.

Footnote:

Just to hit home my usual point one more time, the figure above is ALWAYS a trapezoid, even when sides b and d just happen to be parallel. Just remember that a parallelogram is a special case of a trapezoid!!