Assuming I have a trapezoid where I know the height, bases, and legs, I would like to obtain the width of this trapezoid at any height y. What I want is very similar to the median formula for a trapezoid, except that it needs to work for any height and not just 1/2 height.
For example, suppose I have a trapezoid with height = 10, short base = 10, long base = 20, left leg = 5, right leg = 5. If I want to find the width of the trapezoid at height y = 3, what formula could I use for such a thing?
Best Answer
Say that the bases are of length $b_1,b_2$ (say with $b_1\le b_2$) and that the distance between them is $h$. Then at height (distance from the base of length $b_2$ in the direction directly toward the parallel line through the base of length $b_1$) $y$ we find the width by taking the weighted average of the two bases. In particular, for $0\le y\le h,$ we have the width at height $y$ given by $$w(y)=\frac{y}{h}b_1+\frac{h-y}{h}b_2.$$ It is readily observed by looking at a trapezoid that the width varies linearly with the height, and linear interpolation yields the formula above. As a side note, this formula also works for triangles, taking $b_1=0$ to be the measure of a sort of "one-point base."