The grazing goat on a rope problem and a nearly special triangle

Our Introduction to Geometry book had the classic “goat on a rope” as an example problem. Rather than working through the example, I gave it to my son as a challenge problem and it really gave him fits.

I decided to talk through it with him using our Zometool set and during the talk we discovered that a 3 – 4 – 6 triangle very nearly has a 120 degree angle. Fun little exercise with a surprising geometric result!

First – the problem:


Next – I asked my son to share a few of his initial thoughts about the problem and talk about why it gave him a little trouble. During the conversation here we stumble on the 3-4-6 triangle and think that it might have a 120 degree angle.


Next we went to the table to try to work out the calculation with pencil and paper. My son was a little tired and got tripped up a little by one bit of the calculation, but we did get to the end.

At the end of this piece, we started looking at the 3-4-6 triangle, but decided to go to Wolfram Alpha:


Last up was a quick look at the 3-4-6 triangle on Wolfram Alpha. I probably could have found a way to do this part with the geometry that we already know, but we were both a little tired and investigating the triangle this way seemed just fine.


So, our Zometool set gave us a fun way to look at a pretty standard geometry problem. It is always really fun for me to watch my kids see a problem like this for the first time – I never know where it will go. The extra surprise that came from this particular problem set up was nice, too – a 3 – 4 – 6 triangle nearly has a 120 degree angle!

A fun Zometool story 

Last night we had some neighbors over for dinner at our new house. There were 5 rowdy boys watching a movie in the living room and the adults + one girl were in the kitchen talking and eating pie.

The girl seemed bored of the adult conversation but I saw this Zometool shape catch her eye:

I told her it was from one of our old math projects and brought out the set for her to play around with. She’s in 6th grade, though I don’t know what math classes she’s had, but what she built was awesome.

This was her first shape:

So many great questions to ask with this shape – in fact, I’m going to use it for a project with my kids next weekend. Two of the questions that jumped to my mind were:

(1) What is the area inclosed by the outer hexagon in terms of the area of the inner hexagon?

(2) What is the area of the region between the large and small hexagons in terms of the area of the small hexagon?

I think there’s also a fun opportunity to use this shape to talk about how areas of similar objects scale when the sides scale.

The next shape she made was this:

Here you’ve got an opportunity to talk about how to calculate the area of a trapezoid since her shape shows exactly how two trapezoids can be arranged into a parallelogram. Sort of reminiscent of this recent post from Ben Orlin:

The final shape she built was absolutely awesome – the trapezoids from the previous shape can be rearranged into a pentagon!

This shape has almost endless opportunities and I couldn’t resist showing her a recent project that we did with our Zometool set involving pentagons and Fibonacci numbers since (I think) a similar Fibonacci pattern will emerge from the pentagon she made:

Fibonacci Spirals and Pentagons

So, a super fun evening watching a middle school kid play around with a Zometool set for the first time. It really is incredible what comes out of just playing around with these amazing geometry tools.