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Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have?
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Understanding the New Polyhedron
When we cut off the corners of a cube, we create a new polyhedron with distinct characteristics. Let’s break down the changes in vertices and faces.
Original Cube Properties
- A cube has **8 vertices**.
- A cube has **6 faces**.
Vertices After Cutting Corners
- Each corner (vertex) of the cube is cut off, which transforms it into a triangular face.
- When we cut off the **8 corners**, each corner creates **1 new vertex**.
- Additionally, the original cube’s **8 vertices** will now be replaced by **new vertices** created by the cuts.
**Total New Vertices Calculation:**
- **Original vertices:** 8 (from the cube)
- **New vertices from cuts:** 8 (one from each corner)
- **Total vertices:** 8 (original) + 8 (new) = **16 vertices**.
Faces After Cutting Corners
- The original **6 square faces** of the cube are transformed into **octagonal faces**. Each square face, when the corners are cut, becomes an octagon.
- The **8 triangular faces** are formed from cutting off each corner of the cube.
**Total Faces Calculation:**
- **Octagonal faces:** 6 (one for each face of the cube)
- **Triangular faces:** 8 (one for each corner cut)
- **Total faces:** 6 (octagons) + 8 (triangles) = **14 faces**.
Final Summary
- **Total Vertices:** 16
- **Total Faces:** 14
This new polyhedron is known as a truncated cube. By understanding the changes in vertices and faces, we can appreciate the structure of this fascinating shape!
When we cut off the corners of a cube, we create a new polyhedron with distinct characteristics. Let’s break down the changes in vertices and faces.
Original Cube Properties
- A cube has **8 vertices**.
- A cube has **6 faces**.
Vertices After Cutting Corners
- Each corner (vertex) of the cube is cut off, which transforms it into a triangular face.
- When we cut off the **8 corners**, each corner creates **1 new vertex**.
- Additionally, the original cube’s **8 vertices** will now be replaced by **new vertices** created by the cuts.
**Total New Vertices Calculation:**
- **Original vertices:** 8 (from the cube)
- **New vertices from cuts:** 8 (one from each corner)
- **Total vertices:** 8 (original) + 8 (new) = **16 vertices**.
Faces After Cutting Corners
- The original **6 square faces** of the cube are transformed into **octagonal faces**. Each square face, when the corners are cut, becomes an octagon.
- The **8 triangular faces** are formed from cutting off each corner of the cube.
**Total Faces Calculation:**
- **Octagonal faces:** 6 (one for each face of the cube)
- **Triangular faces:** 8 (one for each corner cut)
- **Total faces:** 6 (octagons) + 8 (triangles) = **14 faces**.
Final Summary
- **Total Vertices:** 16
- **Total Faces:** 14
This new polyhedron is known as a truncated cube. By understanding the changes in vertices and faces, we can appreciate the structure of this fascinating shape!
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Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have?
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Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have? for Class 6 2024 is part of Class 6 preparation. The Question and answers have been prepared according to the Class 6 exam syllabus. Information about Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have? covers all topics & solutions for Class 6 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have?.
Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have? for Class 6 2024 is part of Class 6 preparation. The Question and answers have been prepared according to the Class 6 exam syllabus. Information about Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have? covers all topics & solutions for Class 6 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Each corner of a cube is cut off, leaving a triangular face at each corner and an octagonal face in place of each face of the original cube. How many vertices and faces will the new polyhedron have?.
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