Pizza Party 2

Geometry Level 1

I didn't want a "Home alone" Party, so I invited Jake Lai round to share some pizza. We ordered a pepperoni pizza but when it arrived, we noticed somebody had messed up the cutting. see picture above

Is it still possible for us to have the same amount of pizza?

‣ The pizza has been cut into 8 slices with equal angles of $45°$
‣ Part 1

Yes, we can always have the same amount It depends on which point the pizza is cut through, so i can't tell No, we can never have the same amount if its not cut through the center

4 solutions

Isaac Buckley
Aug 10, 2015

Yes, its possible from any point.

This is fittingly known as the Pizza theorem .

To get an equal amount of pizza we take alternative slices.

This works when we have $4n$ slices for $n\geq 2$ .

Here is a proof without words for this case:

pizzaproof

Click here for an interactive proof without words.

But you may not get the same amount of pepperoni, so a fight could still break out

grabs popcorn

Calvin Lin Staff - 5 years, 10 months ago

And you may not get the same length of the crust. What happens if the pizza was famous for its thin cheesy crust? =D

Wee Xian Bin - 5 years, 10 months ago

Ohhhhh, actually, you do get the same amount of crust.

And that is actually a corollary of the pizza theorem:

With $4n$ slices cut at equal angles around any point of the pizza, alternating slices will lead to an equal distribution.

Do you see how to prove it in 1 line?

Calvin Lin Staff - 5 years, 10 months ago

LOLOLOLOL!

Nihar Mahajan - 5 years, 4 months ago

Practically,, the pizza mfg. can not make a perfect right, Slice cutting accuaracy can not be 100%. it is better to maintain the equality principle maintained at source rather than correction.

Sachin Kapse - 5 years, 9 months ago

Simon Stead
Aug 14, 2015

The pizza theorem is great and would solve this problem in generality, but the far more simple and practical answer is "cut a diameter through the point of intersection" so you have each have half a pizza, with different sized slices.

The question the pizza theorem answers is "can both people have the same amount of pizza in the same sized slices"

Ejun Dean
Aug 22, 2015

A more practical way of thinking about the pizza theorem would be to use the alternate interior angles theorem to show how the slices are equal in area.

Can you explain that proof?

Calvin Lin Staff - 5 years, 4 months ago

Syed Baqir
Aug 25, 2015

In the diagram it shows 8 cuts of pizza so I assumed that each one will get 4 slices since 8/2 = 4 Hence the answer .

Same amount of slices does not mean we get the same amount of pizza.

Isaac Buckley - 5 years, 9 months ago

Then? What is the difference?

Kishore S. Shenoy - 5 years, 9 months ago

One is a volume/area and the other is the number of slices.

Isaac Buckley - 5 years, 9 months ago

@Isaac Buckley – So equal slices means?

Kishore S. Shenoy - 5 years, 9 months ago

Pizza Party 2

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