Here is a wonderful problem I read about in Sanjoy Mahajan’s Street-Fighting Mathematics (link to full book pdf).

Suppose you have an equilateral triangle and you want to cut it in half using a pizza cutter (which can cut curves, not just straight lines). What is the shortest cut you can make?

“Cut in half” means you wind up with two pieces of equal area, not necessarily two identical pieces.

Hint: You might be able to solve this with some fancy calculus, but you can get away with a picture if it’s clever enough.

### Like this:

Like Loading...

*Related*

This entry was posted on October 30, 2010 at 3:45 am and is filed under problems and solutions. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

October 30, 2010 at 4:41 pm

I’d guess an arc with one of the vertixes at the center of the would be circle.

So, let see. Since the suspended angle = pi/3,

1/6 pi r^2 = 1/2 area of triangle.

That give you r.

October 30, 2010 at 5:50 pm

Why do you think the curve is part of a circle?

November 1, 2010 at 1:34 am

I think the problem is equivalent to having a soap bubble wall between two equal amounts of gas in the triangle. And I think the bubble would be circular, and perpendicular to the walls, because that is what bubbles do.

That was easy, if maybe wrong!

November 2, 2010 at 10:42 pm

Lars, that ‘s right, but why is that what bubbles do?

November 2, 2010 at 11:12 pm

[…] problem asked Suppose you have an equilateral triangle and you want to cut it in half using a pizza cutter […]

November 3, 2010 at 7:33 pm

It’s got something to do with tension and symmetry and… I don’t remember the reasoning behind it, but I’ve internalized the result.