Now try to make the above shapes using straws and thread and fill in the table.
| Shapes | Number of faces (F) | Number of corners (V) | Number of edges (E) |
|---|---|---|---|
| Cube/Square Prism | | | |
| Cuboid/Rectangular Prism | | | |
| Triangular Pyramid | | | |
| Square Pyramid | | | |
| Triangular Prism | | | |
Identify any relationship that you may find between the number of faces (F), edges (E), and corners (V). Calculate F + V - E in each case.
Answer (2)
We examined several 3D shapes and calculated the number of faces, vertices, and edges for each. According to Euler's formula, F + V − E = 2 holds true for all the shapes we discussed. This highlights a key relationship in geometry regarding polyhedra.
;
In this task, we will explore some 3D shapes and find a relationship between the number of faces (F), edges (E), and vertices (V), which are also called corners. Specifically, for each shape, we will calculate F + V − E . This relationship is known as Euler's formula for polyhedra.
Let's go through each shape one by one:
Cube/Square Prism :
Number of Faces (F): 6
Number of Vertices (V): 8
Number of Edges (E): 12
Calculate: F + V − E = 6 + 8 − 12 = 2
Cuboid/Rectangular Prism :
Number of Faces (F): 6
Number of Vertices (V): 8
Number of Edges (E): 12
Calculate: F + V − E = 6 + 8 − 12 = 2
Triangular Pyramid :
Number of Faces (F): 4
Number of Vertices (V): 4
Number of Edges (E): 6
Calculate: F + V − E = 4 + 4 − 6 = 2
Square Pyramid :
Number of Faces (F): 5
Number of Vertices (V): 5
Number of Edges (E): 8
Calculate: F + V − E = 5 + 5 − 8 = 2
Triangular Prism :
Number of Faces (F): 5
Number of Vertices (V): 6
Number of Edges (E): 9
Calculate: F + V − E = 5 + 6 − 9 = 2
Upon calculating F + V − E for each shape, we can observe that the result is always 2. This illustrates Euler's formula for polyhedra F + V − E = 2 , which is a fundamental formula in geometry. Euler's formula helps to understand the basic properties of polyhedra and is a useful tool for analyzing 3D shapes.
