Euler's Formula
A convex polyhedron is a polyhedron where a line connecting any two points on the polyhedron that do not lie on the same plane always lies in the interior of the polyhedron (source). The practical realisation and verification of Euler's Formula was an activity that we did with our students on days 6 and 7 of the course. The formula states that, for any convex polyhedron, the sum of the number of vertices and faces is equal to two more than the number of edges. For a quick refresher on vertices, faces and edges, click here. Symbolically, V + F = E + 2.
For example, a tetrahedron (shown below) has four vertices, four faces and six edges which satisfies Euler's Formula.
Using Jodo straws, we asked our kids to work in teams and build as many different polyhedra that they could think of. In keeping with the other activities of the course, this task required kids to move around and actively use their minds and hands. Jayasree Ma'am and I were proud of the teamwork that they displayed in constructing their polyhedra. Some of our kids got quite ambitious with the size of their models too!
Next, we asked each team to count the number of vertices (V), faces (F) and edges (E) of the different solids that they had made. We listed out the values in a tabular format.
Note that we had consciously not mentioned either Euler or his formula up to this point. We wanted our students to study the table that we had filled together and arrive at patterns and conjectures. With a sufficient number of examples, they were able to come up with Euler's Formula. However, they were puzzled by the fact that a few of their solids did not satisfy Euler's Formula. We pushed them to think about how these solids differed from other solids that were obeying the relation.
A polyhedron must have flat polygonal faces, straight edges and sharp corners or vertices. Our children gradually realised that some of their models were contorted and did not satisfy the conditions required for a polyhedron due to the flexibility provided by the straws and connectors of the kit.
To deepen their understanding, we gave them a worksheet with a variety of solids - some polyhedra and others not. We concluded the topic by discussion the Euler characteristic which is the value of (V + F - E) that is equal to 2 for convex polyhedra and not equal to 2 for non-convex polyhedra and solids with holes/deformities.
Next post: Game of Nim
For example, a tetrahedron (shown below) has four vertices, four faces and six edges which satisfies Euler's Formula.
Using Jodo straws, we asked our kids to work in teams and build as many different polyhedra that they could think of. In keeping with the other activities of the course, this task required kids to move around and actively use their minds and hands. Jayasree Ma'am and I were proud of the teamwork that they displayed in constructing their polyhedra. Some of our kids got quite ambitious with the size of their models too!
Next, we asked each team to count the number of vertices (V), faces (F) and edges (E) of the different solids that they had made. We listed out the values in a tabular format.
Note that we had consciously not mentioned either Euler or his formula up to this point. We wanted our students to study the table that we had filled together and arrive at patterns and conjectures. With a sufficient number of examples, they were able to come up with Euler's Formula. However, they were puzzled by the fact that a few of their solids did not satisfy Euler's Formula. We pushed them to think about how these solids differed from other solids that were obeying the relation.
A polyhedron must have flat polygonal faces, straight edges and sharp corners or vertices. Our children gradually realised that some of their models were contorted and did not satisfy the conditions required for a polyhedron due to the flexibility provided by the straws and connectors of the kit.
To deepen their understanding, we gave them a worksheet with a variety of solids - some polyhedra and others not. We concluded the topic by discussion the Euler characteristic which is the value of (V + F - E) that is equal to 2 for convex polyhedra and not equal to 2 for non-convex polyhedra and solids with holes/deformities.
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| The five platonic solids - read more about these fascinating polyhedra here! |
Next post: Game of Nim
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