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Age 14 to 16
Challenge Level

We are still waiting for some good student contributions on this problem but here are some thoughts to help you :

You could start with the three shape posibilities in this illustration :

First the triangle : suppose a length of say 20 cm was folded in the centre, for which angle would the cross-section area be greatest ? If the length had been 40 cm instead of 20 cm would that change your answer ?

Now the rectangle cross-section : maybe start with an overall length of 20 cm again and try different proportions of side length to base. What proportion created the largest area, was that a surprise or an expected result ?

Finally, for now, the arc : for a specified length, say 20 cm, you could curve it a lot or a little. If you curve it a lot you will have more of the full circle but it's a smaller circle. If you curve the material only a little you get less of a larger circle. A spreadsheet might help you find the best choice and ease the burden of calculation.

I hope you enjoy finding that result, if you didn't guess it. Now compare all three results : notice anything ? Can you account for that ?