A follow up to our Tina Cardone geometry project

Had a ton of fun turning a problem that Tina Cardone shared in to a 3D printing project earlier this week:

A Cool Geometry Problem Shared by Tina Cardone

So when I saw this problem in my son’s Geometry book this morning, I couldn’t resist trying a similar project today:

Geometry Problem

The problem is fairly straightforward to state: You go around a square connecting the midpoints of each side to an opposite corner (see the above picture). What is the ratio of the area of the middle square (shaded in the picture) to the area of the original square?

We started our project this morning prior to the videos talking through the problem and actually got all the way to the end. For our project we revisited the problem and then talked about how we could write down the equations of the lines. We need those equations to print the shapes. Our talk this morning lasted about 8 minutes and I broke it up into two pieces here. The first piece is us talking about some basic coordinate geometry and the equations of the horizontal and vertical lines:


The second part of our talk this morning involved writing down the equations of the lines cutting across the square. Turned out to be a nice review of lines, which was one of the things I was hoping to get out of this project:


When I returned from work tonight we revisited the equations of some of the criss-cross lines and eventually wrote down the equations for all four of them. I’m happy for the review of lines that we got in this project – it has been a while since we touched on that subject and a problem about lines from an old math contest gave him a bit of trouble last week, too. Getting all four equations in this part of the project enabled us to tell Mathematica how to draw the shapes we needed to print:


Next up was our little session in Mathematica. Not the best talk we’ve ever done, but we eventually get all of the shapes. As raw as this video is, I think it is a pretty honest description of my nearly non-existent programming skills . . . .


While my son was at his evening karate class I printed the 4 triangles, the 4 quadrilaterals, and the one small square that make up the large square. Here’s a quick shot of the printer finishing up one of the quadrilaterals:


When he got home from karate we explored the problem using the newly printed shapes. I had him go back to our study first to review the original shape. Having been out of the house I didn’t want him trying to reconstruct the original shape wasn’t fresh in his mind. After he built the large square we looked at a different way to see that the small square is 1/5 th of the larger square. In a way, that’s one of the important lessons here – if there’s a geometric result that seems simple, there’s often a fairly simple geometric explanation, too!


So, a super fun project building off of the earlier project from Tina Cardone’s problem. I’m really excited to see ways that our 3D printer can help us with plane geometry – honestly this is an application of 3D printing that I wasn’t really expecting.