# The Trapezoid and the Triangle

Today I was fiddling with some plastic shapes in math class. Two of them caught my eye. A trapezoid and a triangle.

I noticed very quickly that the trapezoid could be visualized as 3 copies of the triangle put together.

Then I noticed something else, something much more interesting. I placed the triangle on the corner of the trapezoid.

Then I rotated the triangle, keeping the bottom left point fixed.

As I did so, I noticed that some of the triangle protruded off the top of the trapezoid. Suddenly I had a question. When is the protruding area largest?

After much work, I found the answer. What surprised me is that I was able to solve the problem using only high school mathematics. I want to share with you some insights I had while solving the problem. But first, here is a digital version of the problem for you to play around with.

http://www.geogebratube.org/student/m79232

Now that you have the idea of the problem, here are some interesting findings. First, the largest area occurs when the angle is equal to 100°. This is weird. I would have expected the largest area to happen at 90°, when the triangle is at a right angle. This is very unexpected to me.

Second, the area does not increase uniformly. Here is a graph of the angle vs the area. As you can see, it takes a long time for the area to start increasing at all. Then it really takes off around 70. It levels off with the highest area at 100, and then sharply dives down after 110. The graph is not symmetric (a mirror image) which is also interesting.

Third, the maximum area occurs when the length of AB is equal to the length of CD. This makes intuitive sense because the base of the protrusion is “centred” on the top of the trapezoid.

Overall, this was a fun and engaging investigation. Now I need to figure out how to implement it in my classroom…

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2 Comments

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### 2 responses to “The Trapezoid and the Triangle”

1. Scrolls Apprentice

Awesomesauce. First of all, I didn’t even know about GeoGebraTube… swell! Second, I’d be interested to know if the area of the other protruding section achieves its maximum area at the same angle (100) or if it too, is most when it is centred. Another question, at which angle is the combined area of the protruding edges the most?

• Great math problems spawn further questions. I’m glad I could pique your interest 🙂

The answer to the area of the second protruding section is yes. It achieves the same maximum area but its at 180-100 = 80 degrees.

As to the question about the maximum combined area, I’ll leave you to figure that out 😉