Question

Check that the numbers of vertices, edges, and faces of a cube are equal respectively to the numbers of faces, edges and vertices of an octahedron.

Short answer

The cube and octahedron correspond with 6 faces/vertices, 12 edges, and 8 vertices/faces.

Step-by-step solution

Recall Properties of a Cube

A cube is a three-dimensional shape. It has 6 faces, 12 edges, and 8 vertices.

Recall Properties of an Octahedron

An octahedron is also a three-dimensional shape. It has 8 faces, 12 edges, and 6 vertices.

Compare Corresponding Properties

Identify the correspondence: the number of cube's vertices (8) matches the number of octahedron's faces (8), the number of cube's edges (12) matches the number of octahedron's edges (12), and the number of cube's faces (6) matches the number of octahedron's vertices (6).

Conclude the Comparison

Based on the comparison, each property of the cube perfectly matches a corresponding property of the octahedron, confirming the validity of the statement.

Key concepts

Cube

A cube is one of the most recognizable three-dimensional geometric solids. It has six equally sized square faces, which makes it a regular polyhedron. Each face of the cube meets another at a right angle, forming twelve edges in total. Every corner of a cube, where the edges meet, is called a vertex. In total, a cube has eight vertices. Think of a classic dice as a great physical representation of a cube. Here's a quick recap of the key features of a cube:

• 6 square faces
• 12 edges
• 8 vertices

The relationship between these features is essential in understanding the structure of the cube and, when compared to other shapes, can reveal intriguing properties such as those seen between a cube and an octahedron.

Octahedron

An octahedron is a fascinating polyhedron that may not be as immediately recognizable as a cube but holds its own set of intriguing geometric properties. Imagine two pyramids base-to-base; that's what an octahedron looks like. It consists of eight triangular faces, twelve edges, and six vertices. The symmetry of an octahedron is such that its appearance remains unchanged no matter how you rotate it. Here’s a summary of an octahedron’s features:

• 8 triangular faces
• 12 edges
• 6 vertices

Each of its triangular faces is an equilateral triangle, making it another example of a regular polyhedron. Understanding the structure and properties of an octahedron can provide deep insights into geometric relationships, especially when compared to other solids like the cube.

Vertices-Edges-Faces Relationship

The relationship between vertices, edges, and faces in geometric solids is a profound area of study in geometry. This relationship can often be described using Euler's formula for polyhedra, which states: \[ V - E + F = 2 \]where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces. For the cube:

• \( V = 8 \)
• \( E = 12 \)
• \( F = 6 \)

Substituting these into Euler's formula gives \( 8 - 12 + 6 = 2 \), satisfying the formula. Now, considering an octahedron:

• \( V = 6 \)
• \( E = 12 \)
• \( F = 8 \)

Again, Euler's formula holds: \( 6 - 12 + 8 = 2 \). This remarkable consistency illustrates the inherent elegance in geometric solids. Moreover, examining the quantity equality between these two shapes reveals that each property (vertices, edges, faces) of the cube perfectly corresponds with a different property of the octahedron.