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Archimedes Could Solve This Problem!

M.J. (Bellingham, WA) shared this problem with me this summer....

You have a rectangle....no dimensions given.

You have several blue triangles inside the rectangle, arranged so that their bases exactly cover the base of the rectangle. In this diagram, it is four triangles but it could be more or less.

Question: The cumulative area of the triangles equals what percentage of the area of the rectangle?

Extension Question: Suppose the situation was three-dimensional, with a rectangular box containing pyramids. In an analogous manner, the square-bases of the pyramids exactly cover the base of the box and the apexes of the pyramids are all points on the "ceiling" of the box. The cumulative volume of the pyramids equals what percentage of the volume of the rectangular box?

Note: Investigate why Archimedes would be able to solve this problem.

Hint: Think...you do not know any of the dimensions....of either the triangles of the rectangle, but you do know something important about both the triangles and the rectangle!

Also, what if you tried to recreate this problem on a geoboard...what could you do?

Solution Commentary: Visualize the problem as blue curtains hanging by hooks on a curtain rod over a rectangular window. Without changing the base segments of the triangles, imagine sliding all of the top hooks over to the right hand corner of the rectangle. As you slide the hooks, why does the area of each triangle remain constant? And, what does the final arrangment look like....if you can visualize this, you know the answer to the problem.