### Problem Statement **5)** For a regular hexahedron, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler's equation for that polyhedron. ### Explanation A regular hexahedron is another term for a cube. A cube has: - **Number of Faces (F)**: 6 - **Number of Vertices (V)**: 8 - **Number of Edges (E)**: 12 Euler's formula for polyhedra states that for any convex polyhedron: \[ V - E + F = 2 \] Let's verify Euler's equation for the cube: Given: - \( V = 8 \) - \( E = 12 \) - \( F = 6 \) Substitute these values into Euler's formula: \[ V - E + F = 8 - 12 + 6 = 2 \] Therefore, Euler's equation holds true for a regular hexahedron (cube).
### Problem Statement **5)** For a regular hexahedron, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler's equation for that polyhedron. ### Explanation A regular hexahedron is another term for a cube. A cube has: - **Number of Faces (F)**: 6 - **Number of Vertices (V)**: 8 - **Number of Edges (E)**: 12 Euler's formula for polyhedra states that for any convex polyhedron: \[ V - E + F = 2 \] Let's verify Euler's equation for the cube: Given: - \( V = 8 \) - \( E = 12 \) - \( F = 6 \) Substitute these values into Euler's formula: \[ V - E + F = 8 - 12 + 6 = 2 \] Therefore, Euler's equation holds true for a regular hexahedron (cube).
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Problem Statement
**5)** For a regular hexahedron, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler's equation for that polyhedron.
### Explanation
A regular hexahedron is another term for a cube. A cube has:
- **Number of Faces (F)**: 6
- **Number of Vertices (V)**: 8
- **Number of Edges (E)**: 12
Euler's formula for polyhedra states that for any convex polyhedron:
\[ V - E + F = 2 \]
Let's verify Euler's equation for the cube:
Given:
- \( V = 8 \)
- \( E = 12 \)
- \( F = 6 \)
Substitute these values into Euler's formula:
\[ V - E + F = 8 - 12 + 6 = 2 \]
Therefore, Euler's equation holds true for a regular hexahedron (cube).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55a87db1-34dd-4361-bb49-0fd4dde658a5%2F83cf95e3-f146-4274-9b84-09198baaa4d1%2Fbzf3dy.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**5)** For a regular hexahedron, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler's equation for that polyhedron.
### Explanation
A regular hexahedron is another term for a cube. A cube has:
- **Number of Faces (F)**: 6
- **Number of Vertices (V)**: 8
- **Number of Edges (E)**: 12
Euler's formula for polyhedra states that for any convex polyhedron:
\[ V - E + F = 2 \]
Let's verify Euler's equation for the cube:
Given:
- \( V = 8 \)
- \( E = 12 \)
- \( F = 6 \)
Substitute these values into Euler's formula:
\[ V - E + F = 8 - 12 + 6 = 2 \]
Therefore, Euler's equation holds true for a regular hexahedron (cube).
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