An old James Tanton-inspired 3d printing project pays off today!

Last summer James Tanton posted this problem on Twitter:

What is surface area of figure formed by all points within a distance 1 from a regular tetrahedron with faces of area 1?

— James Tanton (@jamestanton) June 20, 2014

That problem led to a super fun 3D printing project with the boys:

James Tanton’s Geometry Problem and 3D Printing

To my great (and wonderful) surprise, today my older son and I came across a similar problem from an old AMC 10:

Problem #17 from the 2008 AMC 10A hosted at Art of Problem Solving

Here’s the problem:

“An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle but not more than 3 units from a point of the triangle?”

Of course we had to dig out our old 3d printed objects to start the problem, and then my son drew in the analogous 2 dimensional figures:

 

Next we moved to a second sheet of paper and my son solved the problem. I like his “inclusion / exclusion” way of counting the area here, though I understand that some people would probably prefer a more geometric approach.

 

How fun to have one of our old 3d printing projects help us solve a challenging AMC 10 problem!