What is the volume of a pizza?

This little 'puzzle' has caused me a fair bit of enjoyment, and it has pleased a few colleagues and daughter #2 (hopefully son #1, daughter #1 and wife #1 will read this blog, and enjoy the puzzle too). The puzzle isn't original, of course, but was seen in a tweet by QI Elves.

At work, I have a few colleagues who like me are mathematicians; some of whom I have shown this to. But on my last day before work I was reviewing some work with two colleagues, neither of whom were mathematically minded: the work was easy, it was the last day, we had time: I asked the more junior colleague to tell me something interesting from the client project (nothing: alas, it was entirely trivial): she gave as good as she was given, asked me to tell her something interesting, and got her own ready (she always comes to meeting with pen and notebook, to write interesting things down, or to write to do lists down); so I borrowed her notebook, and drew the following pizza.

What is the volume of a pizza, radius z, depth a?

(no, it is not a trick question, and no, it doesn't depend on whether it is deep pan or not).

The requirement is also to simplify the expression.

Answer

Most people, including I am pleased to say daughter #2, so private education has paid off) get to the answer fairly easily, sometimes having to scratch the depths of long forgotten maths lessons. The more mathematically minded struggle with it, thinking it is too easy, thinking there must be a catch.

The most common answer is π r^2 a, until I remind people that the radius is not r, but z. So most people get to π z^2 a, which is a good enough answer.

But of course, the question was to simplify it. z^2 can be rewritten as zz, and π can be written as pi, so the expression simplifies to:

pizza : the volume of the pizza is pizza.

QED, as we say here in the Maldives. Now, what is for lunch?

Pro-tip

I am not sure that quinquagenarian's should give iPad pro-tips, but since it is my blog, and I can if I want to, the way I got π was to add the Greek keyboard to my iPad. Under settings, keyboards, the iPad permits several keyboards at once.

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