Pyramid problems.

Problem. Given a triangular base and a given volume, find a pyramid with the given base and given volume which has minimum side area. If possible determine if the solution is unique.

Discussion. Actually, this problem is solved nicely using a Lagrange muliplier towards the end of Lec 13: Lagrange multipliers | MIT 18.02 Multivariable Calculus, Fall 2007. Find it on YouTube. Briefly, you express the side area $A$ as a sum $\sqrt{h_i^2+h^2}\,l_i$ where $h$ is the height of the pyramid and $l_i$ is the length of the $i^{th}$ side of the pyramid (the side opposite the $i^{th}$ vertex) and $h_i$ is the distance from base of the top of the pyramid to the $i^{th}$ side. It turns out that the minimum side area is achieved which $h_1=h_2=h_3$, that is when the top sits directly over the incenter of the triangle. Since it is known that the incenter is the weighted average of the vertices of the triangle where the weights are the lengths of the opposite sides, the minimum area can be computed. Below is an interact which computes this for a given triangle and given height.

We graph the area function by expressing it as a function of the point $(x,y)$ sitting directly over the top of the pyramid. That way we can verify visually that the minimum side does occur when $(x,y)$ is the incenter.

In the code below, we assume the triangle has base vertices $(0,0,0)$ and $(100,0,0)$. You choose in data [a,b,h] the top vertex $(a,b,0)$ of the triangle and the height $h$ of the pyramid. Then it computes the incenter $(x_1,y_1)$ of the base and the side area $g(x,y)$ of the pyramid when the top of the pyramid is above $(x,y,0)$. Finally it graphs $g(x,y)-g(x_1,y_1)$, the side area of the pyramid reduced by the side area of the pyramid with top $(x_1,y_1,h)$.
Press the Evaluate button and expand the cell by pressing the upper right hand corner of the cell.

Here we have just the interact without access to the code.

Some questions that occur about this situation.

1. Under what condition would the side area function $Ar(x,y)$ be nearly constant when $(x,y)$ lies in the base of the pyramid?
2. Why does the area function grow outside the base of the pyramid?
3. Suppose the pyramid has a rectangular base. Where do you think the minimum side area would occur?
4. What about other types of quadrilaterals such as parallograms, trapezoids, cyclic quadrilaterals?

Here is some work on that last question: Can we find a closed form solution to the problem of finding the minimum of the side area function for a pyramid with a given quadrilateral base and given height?

What we have is here an interact which graphs the side area of a pyramid with a quadrilateral base $(0,0,0),(10,0,0),(a,b,0),(c,d,0)$ and a fixed height $h$.

1. Under what condition would the side area function $Ar(x,y)$ be nearly constant when $(x,y)$ lies in the base of the pyramid?
2. Why does the area function grow outside the base of the pyramid?
3. The interact computes the minimum using a numerical method, starting from an initial guess $(x1,y1)$. Is there a symbolic method here as there was in the triangle base case?