1. Space Figures and Cross Sections
Lesson 11-1
Check Skills You’ll Need
(For help, go to Lesson 1-3.)
For each exercise, make a copy of the cube
at the right. Shade the plane that contains
the indicated points.
1. A, B, and C
2. A, C, and G
3. F, D, and G
4. the midpoints of AD CD, EH, and GH
Check Skills You’ll Need
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2. Space Figures and Cross Sections
Lesson 11-1
Homework
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3. Space Figures and Cross Sections
Lesson 11-1
Notes
A polyhedron is a three-dimensional figure whose surfaces are polygons. Each polygon is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more edges.
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4. Space Figures and Cross Sections
Lesson 11-1
Additional Examples
Identifying Vertices, Edges and Faces
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
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5. Space Figures and Cross Sections
Lesson 11-1
Notes
Euler is pronounced “Oiler.”
Reading Math
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6. Space Figures and Cross Sections
Lesson 11-1
Additional Examples
Using Euler’s Formula
Use Euler’s Formula to find the number of edges on a solid 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
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7. Space Figures and Cross Sections
Lesson 11-1
Notes
A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure.
To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face.
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8. Space Figures and Cross Sections
Lesson 11-1
Additional Examples
Verifying Euler’s Formula
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
= 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 Euler’s Formula in two dimensions
= Substitute.
Quick Check
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9. Space Figures and Cross Sections
Lesson 11-1
Notes
A cross section is the intersection of a three-dimensional figure and a plane.
The cross section is a triangle.
The cross section is a rectangle.
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10. Space Figures and Cross Sections
Lesson 11-1
Additional Examples
Describing a Cross Section
Describe this cross section.
The plane is parallel to the triangular base of the figure, so the cross section is also a triangle.
Quick Check
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11. Space Figures and Cross Sections
Lesson 11-1
Additional Examples
Drawing a Cross Section
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
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12. Space Figures and Cross Sections
Lesson 11-1
Lesson Quiz
1. Draw a net for the figure.
Sample:
Use Euler’s Formula to solve.
2. A polyhedron with 12 vertices and 30 edges has how many faces? 20
3. A polyhedron with 2 octagonal faces and 8 rectangular faces has how many vertices?
4. Describe the cross section.
5. Draw and describe a cross section formed by a vertical plane cutting the left and back faces of a cube. 16
Circle
Check students’ drawings; rectangle.
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13. Space Figures and Cross Sections
Lesson 11-1
Check Skills You’ll Need
(For help, go to Lesson 1-3.)
For each exercise, make a copy of the cube
at the right. Shade the plane that contains
the indicated points.
1. A, B, and C A, B, and G
3. A, C, and G A, D, and G
5. F, D, and G B, D, and G
7. the midpoints of AD CD, EH, and GH
Original
Check Skills You’ll Need
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14. Space Figures and Cross Sections
Lesson 11-1
Check Skills You’ll Need
Solutions 7.
Original
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