Transcript Space Figures and Cross Sections LESSON 11-1 Additional Examples How many vertices, edges, and faces of the polyhedron are there? List them. There are 10

Space Figures and Cross Sections
LESSON 11-1
Additional Examples
How many vertices, edges, and faces of the
polyhedron are there? List them.
There are 10 vertices:
A, B, C, D, E, F, G, H, I, and J.
There are 15 edges:
AF, BG, CH, DI, EJ, AB, BC, CD,
DE, EA, FG, GH, HI, IJ, and JF.
There are 7 faces:
pentagons: ABCDE and FGHIJ, and
quadrilaterals: ABGF, BCHG, CDIH, DEJI, and EAFJ
Quick Check
HELP
GEOMETRY
Space Figures and Cross Sections
LESSON 11-1
Additional Examples
Use Euler’s Formula to find the number of edges of a
polyhedron with 6 faces and 8 vertices.
F+V=E+2
Euler’s Formula
6+8=E+2
Substitute the number of faces and vertices.
12 = E
Simplify.
A solid with 6 faces and 8 vertices has 12 edges.
Quick Check
HELP
GEOMETRY
Space Figures and Cross Sections
LESSON 11-1
Additional Examples
Use the pentagonal prism from Example 1 to verify
Euler’s Formula. Then draw a net for the figure and verify
Euler’s Formula for the two-dimensional figure.
Use the faces F = 7, vertices V = 10, and edges E = 15.
F+V=E+2
Euler’s Formula
7 + 10 = 15 + 2
Substitute the number of faces and vertices.
Draw a net.
Count the regions: F = 7
Count the vertices: V = 18
Count the segments: E = 24
F+V=E+1
Euler’s Formula in two dimensions
7 + 18 = 24 + 1 Substitute.
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HELP
GEOMETRY
Space Figures and Cross Sections
LESSON 11-1
Additional Examples
Describe this cross section.
The plane is parallel to the triangular base of the figure, so the cross
section is also a triangle.
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HELP
GEOMETRY
Space Figures and Cross Sections
LESSON 11-1
Additional Examples
Draw and describe a cross section formed by a vertical plane
intersecting the top and bottom faces of a cube.
If the vertical plane is parallel
to opposite faces, the cross
section is a square.
Sample: If the vertical plane is
not parallel to opposite faces,
the cross section is a rectangle.
Quick Check
HELP
GEOMETRY