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Carpentry and fencebuilding problems

Katherine and I are back in Seattle after a summer away, exploring the mathematics of  activities not normally associated with math: building a kitchen, planting gardens, and putting up goat fences. Of course, these activities were sometimes more mathematical than they looked. I used more trig than I have in years to make sure all the cuts on the miter saw were made at the correct angles.

Now we’re back, and beginning a new academic year. Our classes are off to a rollicking start, and we’re leading workshops for math teachers in schools around Seattle, which is a real pleasure. It’s sad to be away from the country, but good to be back in the city too.

In the spirit of our summertime activities, here are a few math puzzles to get the year started:

1. Squares have a way of becoming rhombuses if you’re not careful, and when you’re setting the walls onto the foundation, you have to make sure that your four perfectly-cut, 12-foot-long two by fours actually meet at right angles, forming a square, rather than skewing off and becoming a non-rectangular parallelogram. Fortunately, carpenters have a simple way to check that the wood is square by making just two measurements with a tape measure. Can you figure out how they do it?

2. Of all rectangles, the square is the most “efficient,” in the sense that if you have a certain amount of fence, and form it into a square, you’ll have a larger garden than if you form it into any other size rectangle. However, that fact goes out the window if you are fencing your garden against a preexisting wall. What rectangle is most efficient when you have the wall?

Have answers? Questions? Generalizations? Drop them in the comments and I’ll reveal my answers to these questions later.

 

Comments 2

  1. Michael Paul Goldenberg

    Nice little pragmatic question. Not being a craftsman of any sort, I have to resort to geometry and would presume that the key is having the diagonals equal, which would not be true for a rhombus. Since you know the sides are equal by previous measurement (when you cut them), you’ve got the property that the sides and the diagonals are equal leading to the conclusion that the angles are equal and all 90 degrees.

    1. Post
      Author
      Dan

      You got it, Michael. In the course of building this kitchen, we ended up measuring the diagonals of various squares and rectangles many, many times.

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