One Son has his A.S. Engineering from the nearby junior college and is transferring to a state school to finish his B.S.
One daughter is a Dual Bio/Comp Sci major with Chem minor.

The are both beyond the year of Calc I had as an undergrad.  I was a Bio/Chem undergrad.

Other than some complicated statistics, they are pretty smug about getting beyond me in math.
I am a touch smarter than one of them.  One of them is a touch smarter than me.  They each know they are dwelving into specific areas of education further than I have gone and are getting cocky.

When both are home they occasionally team up and beat up on me.  I am totally cool with it and enjoy it.

Hey Dad, are you sure your putting the right amount of   In the pond?
Hey Dad, what is the max pasture you can enclose with that amount of fencing?
Hey Dad, if you are adding two inches of sand to the dressage arena how much will it weigh?
Hey Dad, if it gets to ten below this winter would those fish survive?

Fair enough.

I usually fire right back at them.
Yes, if we get the average rainfall.  
Easy, 1000 ft for that one, or 1500 ft for that one, as the circumference of a circle and the area of that circle will be the max.
Wet or dry?
Yes, if they stay in the bottom of the deep portion, they would die in the shallow part.

They nailed me with another one.
What is the minimum area you could fence in with 1000 or 1500 ft of fencing.  I fired back, zero square feet.  I could just run two lines of fence enclosing nothing.

They have followed up with:
What is the average area you could fence in with 1000 sf or 1500sf of fence.

I responded with half the area of a circle with those respective circumferences.
Based on a circle being the max, and the ability to line up the fencing with zero sf.

They have teamed back up and am asking me if I am sure.

Uh Oh.
What have they caught me with?

Is there a specific shape representing the average sf enclosed by a perimeter,
Or some civil engineering or other standard calculation that determines this?

Based on the fact nobody would ever spend a bunch of money to utilize their fencing in a manner to enclose zero square feet.

Great that they are learning and thinking. You have done well as a father.

Heck, I want my kids to be smarter than me and wealthier and happier and more advanced in their relationships with God.

What's the old saying by the one time head of Mensa - "If I (or you) have to be the smartest one in the room, you have to keep the room really small (because there is always someone smarter in some way)."

The smallest size is zero, the largest size is the circle, all sizes in between are possible. The average of the sizes is clearly half the maximum. You can state you are sure.

If instead you wish to average over all the possible ways to arrange such a fence, the answer isn't so clear, and it seems likely not enough information has yet been given to solve the problem. For a similar example, consider the case where you have four edges to place on an infinite checkerboard. There are an infinite number of ways to fence in a single square, and infinitely more ways to fail to do so, so the average area enclosed by all possible edge placements will be zero.

You should challenge them back with: A farmer has a square tract of land and he will give you a piece of it for free, the catch is you have to pay him to fence it off for you at $X per foot of fence line. What shape gets you the best $/acre?

Depending on what type of engineering he pursues he may go way past you.

Electrical (and Chemical and some other corners) lives by the Laplace transform and reduces Differential Equations to algebra in that domain.

Put a fence around yourself and transform it so the farmer is inside.
You are now on the outside and take everything else.

Since you have Hewlett-Packard's 15C calculator in your title, check this out:

https://www.swissmicros.com

Imitation is a form of flattery.

those are very cool