LON-CAPA Euler's Formula for Polyhedra

Browsing resource, all submissions are temporary.

,

Euler's Formula

One of Euler's formulas pertains to solid objects called "polyhedra" (from the Greek for "many faces"). It states that the number of faces (F) plus the number of corners (C) minus the number of edges (E) of any polyhedron is equal to the "Euler characteristic", the quantity 2. In algebraic notation:

F + C - E = 2

For example, examine the two figures below, a cube and a tetrahedron. (You can click in the figure and move your cursor to rotate the object).

	Cube	Tetrahedron

Faces (F)	6	4
Edges (E)	12	6
Corners (C)	8	4
Euler's characteristic (F+C-E)	2	2

Now see if his formula applies to the following polyhedra.
Click on the name of the polyhedron to open a window showing its shape.
Count the faces (F), corners (C), and edges (E) and enter your values in the blanks.
(The faces are colored to make it easier to keep track of them).
The "New Problem Variation" button at the top will erase the answers and reset the problem.

Octahedron
F = C = E = F+C-E =

Tries [_1]

Dodecahedron
F = C = E = F+C-E =

Tries [_1]

Icosahedron
F = C = E = F+C-E =

Tries [_1]

Truncated Icosahedron
F = C = E = F+C-E =

Tries [_1]