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#249 Espresso geometry

Set by Katie Steckles

The Circle Coffee Shop was getting a rebrand and its big, circular logo on the shop wall was getting repainted to match its new identity, The Equilateral Triangle Coffee Shop. The painter was just packing up their brushes when the coffee shop boss came stomping over.

鈥淭his isn鈥檛 right! You鈥檝e painted an equilateral triangle that fits exactly inside the old circle logo! We wanted the new triangle logo to just cover the whole of the old one!鈥� The painter looked doubtful. 鈥淚鈥檝e just used a whole tin of paint doing this. If you want more paint on the wall, you鈥檒l have to pay for any extra!鈥�

Exactly how much more paint will the coffee shop boss have to stump up for?

 

Solution next week

#248 Pizza parity

Solution

Call the area of the pepperoni pizza P and that of the broccoli pizza B. The total area of the pizzas is P + B. Any line that passes through the centre of a pizza is its diameter, so the line L (black dashed line) that joins the two centres will cut both pizzas in half. Poppy cuts along L up to the edge of B, then continues around B until she can rejoin L (red dashed line). The larger piece of pepperoni pizza now has area P/2 + B/2, while the smaller bit has P/2 鈥� B/2. Add the whole of the broccoli pizza to the smaller bit of pepperoni pizza, and both portions are now P/2 + B/2.

Line L through centre of both pizzas