A net is a pattern made when the surface of a 3-D figure is laid out flat, showing each face of the figure. A solid may have different nets.
Verification of Euler's Formula for the Polyhedron: 7+7=12+2 Net:
Verification of Euler's Formula for the Net: 7+12=18+1
Practice makes perfect
We will verify Euler's Formula for the given polyhedron. Then, we will draw the corresponding net and verify Euler's Formula for the 2-D shape.
Verifying Euler's Formula for the Polyhedron
Let's consider the given 3-D figure.
We can see that this polyhedron has 7 faces, 7 vertices, and 12 edges. Euler's Formula states that the sum of the number of faces F and vertices V of a polyhedron is two more than the number of its edges E. Let's verify this!
A net is a pattern made when the surface of a 3-D figure is laid out flat showing each face of the figure. A solid may have different nets.
The variables F, V, and E are used for different elements in 3-D and 2-D shapes.
Variable
In 3-D
In 2-D
F
Faces
Regions
V
Vertices
Vertices
E
Edges
Segments
Let's draw a net for the given figure.
Verifying Euler's Formula for the Net
In two dimensions, Euler's Formula states that the sum of the number of regions F and vertices V is one more than the number of segments E. The net we drew has 7 regions, 12 vertices, and 18 segments. Let's verify Euler's formula for our net!
