New Problem: Halving the Triangle

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.

6 Responses to “New Problem: Halving the Triangle”

1. Nik Says:

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.

2. Mark Eichenlaub Says:

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

3. Lars P Says:

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!

4. Mark Eichenlaub Says:

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

5. Solution: Halving the Triangle « Arcsecond Says:

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

6. Lars P Says:

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.