“How will this help me in my later life?”

While many high school math teachers consider this to be an annoying question, they’re wrong. Far more annoying are the math teachers who can’t produce a good answer.

Case in point: It takes a modest background in mathematics to appreciate the nuances of the surface area to volume ratio.

I mentioned this ratio a couple of weeks ago (“The mathematics of organizational dysfunction,“Keep the Joint Running, January 22, 2007), describing the difference between enclosing a volume in one big balloon and a bunch of small ones. The small ones, you’ll recall, require a lot more surface area, which means their contents have better contact with the world around them.

Another way to provide more exposure is to squash a volume flat. Compare the surface area of a one meter cube (6 square meters) to that of the same volume mashed down to one millimeter thick (a bit more than 2,000 square meters).

In the flatter solid, each bit of volume has much more access to the outside world.

When will you use this information in your later life? Perhaps tomorrow, when you take a fresh look at the organization you manage. Because you have to decide how much surface area you want it to have. It’s a tough decision.

Flattening the organization has solid benefits, among them a decrease in the distance between employees and vendors, customers, and the company’s top executives.

On the other hand, visualize the flat solid and you might think employees would find themselves farther apart from each other than in the cube. In practice it generally works out to be another advantage. Here’s why:

Imagine a company with a typical five-layer hierarchy. Theory says that for any employee to reach any other employee, the biggest distance is nine steps — supervisor to manager, to middle manager, to executive to CEO and back down the chain again. If the organization sheds just one layer of management, that shrinks to seven steps (supervisor to manager to executive to CEO and back).

In practice, the distance between employees shrinks even more. That’s because in a flat organization, managers and executives have too little time to insist that employees respect the chain of command. Go back to the five-layer company and imagine it employs 5,000 people. Each manager has, on the average, 5.49 direct reports (the fifth root of 5,000).

Cut out just one layer. Now, each manager has 8.41 direct reports (the fourth root). Cut out another layer and each manager has 17.10 direct reports (the third root). Put yourself in this position — do you have time to shuttle requests for introductions back and forth? I don’t think so. You’ll tell employees wanting to talk to each other to just pick up the phone and stop bothering you.

So mathematically speaking, flat organizations have to be more informal than more hierarchical ones. That’s why executives who figure their companies are too bureaucratic typically fire a bunch of middle managers.

Flat organizations sure seem attractive. They are, in fact, quite popular among consultants who generally need to find excuses to lay off employees anyway. Middle managers have a hard time demonstrating the value they provide under the best of circumstances, so the advantages to be had from flattening the hierarchy can be irresistible.

If only the picture was this one sided. Flat organizations have three serious disadvantages, too. The first is that nobody has time to give employees much attention. That translates to managers engaging in send-and-forget delegation, because no manager has the time to meet with employees often enough to find out how their assignments are progressing. They have no time to keep track, let alone to provide informed guidance.

The second disadvantage is that middle management is the training ground for executives. Cut out a layer and companies find themselves either promoting people who are too green for their assignments, or, lacking qualified internal candidates, having to hire from the outside (which doesn’t pass the Mom test: What if everyone did it?)

And finally, there’s this disadvantage: Flatter organizations have more people competing for fewer opportunities in the next layer up — a near-perfect formula for building a nasty political environment.

To flatten, then, or not to flatten. The decision depends on what problems you’re trying to solve, and which ones you’re willing to live with.

Do the math.