Ceptimus was hungry,....very hungry.
Fortunately he did have a bit of cash and his route home from the 8th International Geometry Congress did pass a local Pizzeria.
He walked in and sat down, delighted to read that they had on a "special offer" of one large pizza for the price of a small and medium combined. Keen to get value for money, he wondered if the special deal was such a bargain after all. Easy to decide, he thought, I'll just measure their respective radii and calculate the areas.
Reaching in his jacket pocket, he didn't find his trusty tape measure, however, but a plastic protractor he had picked up from one of the advertiser's stands at the congress.
Never mind, he thought, this will do just as well. I'll ask the waiter if he can cut the pizzas in half and then using only my protractor I can easily decide which deal is the better one.
What did Ceptimus do?
Pizza problem.

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He takes one half from each of the three pizzas and arranges the cut sides into a trianlge. He uses the protractor to measure the angles of the triangle. Then, using a bunch of trigonometry that I've completely forgotton, he calculates their relative areas.
Or.. he uses the little ruler that is often found on the straight edge of many plastic protractors and measures them!
Or.. he uses the little ruler that is often found on the straight edge of many plastic protractors and measures them!

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I'm going with Whim's solution, and filling in the blanks.
<table bgcolor="white"><tr><td>The arrangement that Whim describes should be familiar to those who have seen the geometrical proof of the Pythagorean theorem, except with semicircles instead of squares.
The semicircles will have an area in the same proportion to their containing squares, no matter their size, so the Pythagorean relation (area S + area M = area L) still holds if the pizzas form a right triangle.
So, if the areas are the same, then the angle between the small and medium pizzas must be 90 degrees. If the area of the large pizza is greater, then the angle will be greater than 90 and if it's smaller, then the angle will be less than 90.</td></tr></table>
Of course, after taking his protracted measurement, the waiter insists that Ceptimus pay for all three.
James (Eating pizza while typing this solution.)
<table bgcolor="white"><tr><td>The arrangement that Whim describes should be familiar to those who have seen the geometrical proof of the Pythagorean theorem, except with semicircles instead of squares.
The semicircles will have an area in the same proportion to their containing squares, no matter their size, so the Pythagorean relation (area S + area M = area L) still holds if the pizzas form a right triangle.
So, if the areas are the same, then the angle between the small and medium pizzas must be 90 degrees. If the area of the large pizza is greater, then the angle will be greater than 90 and if it's smaller, then the angle will be less than 90.</td></tr></table>
Of course, after taking his protracted measurement, the waiter insists that Ceptimus pay for all three.
James (Eating pizza while typing this solution.)
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