Question

The number of hexagonal faces that are present in a truncated octahedron is

Short answer

There are 8 hexagonal faces in a truncated octahedron.

Step-by-step solution

Understanding the Truncated Octahedron

A truncated octahedron is a polyhedron resulting from truncating (cutting off) a regular octahedron. Each of the eight vertices of an octahedron is cut off, resulting in a shape that has squares and hexagons for faces.

Identifying the Hexagonal Faces

After truncation, each original vertex of the octahedron becomes a hexagonal face. Since a regular octahedron has eight vertices, the truncated octahedron would have eight hexagonal faces.

Counting the Hexagonal Faces

Simply count the number of vertices on the original octahedron, which is equivalent to the number of hexagonal faces on the truncated octahedron.

Key concepts

Polyhedron

When it comes to understanding complex geometric shapes, starting with the basics is essential. A polyhedron is a three-dimensional shape with flat faces, straight edges, and sharp corners or vertices. Imagine a polyhedron like a puzzle; each piece is a flat surface that fits together with others to create a closed three-dimensional figure. Polyhedra are categorized by the shapes of their faces and by how many faces, edges, and vertices they have.

Each face of a polyhedron is a polygon, which is a 2D shape with straight sides. When all its faces are regular polygons and the same number of faces meet at each vertex, a polyhedron is called 'regular,' like the traditional cube or tetrahedron. However, when we move to complex figures, such as the truncated octahedron in our exercise, the rules change slightly. A truncated octahedron is considered a semi-regular or Archimedean solid because it is made up of two or more types of regular polygons meeting in identical vertices.

Geometric Solids

Geometric solids are three-dimensional figures, like spheres, cubes, and cylinders that we see in our everyday world. They are defined by their volume, surface area, and the curves or flat surfaces that make up their boundaries. Geometric solids have depth, width, and height, differentiating them from two-dimensional shapes which only have length and width.

To deepen our grasp of these solids, we create models or use formulas to calculate properties such as volume, which tells us how much space the solid occupies, or surface area, which gives us the total area of all the surfaces of the solid. In the case of the truncated octahedron from our exercise, understanding geometric solids helps us to appreciate the transformation from a regular octahedron, which has eight triangle faces, into a more complex shape with eight hexagonal and six square faces.

Hexagonal Faces

A crucial aspect of many polyhedra, including our truncated octahedron, is the presence of hexagonal faces. A hexagon is a six-sided polygon, and when it's regular, all sides and angles are equal. In three-dimensional geometry, hexagons add fascinating properties to solids.

In our truncated octahedron example, the original octahedron's vertices are cut off or truncated, transforming them into hexagonal faces. This results in a total of eight hexagonal faces, which neatly tessellate, or fit together without gaps, alongside the other faces. This characteristic is vital for students to visualize the transition of shapes through truncation and helps anchor the concept of altered vertex figures in the study of polyhedra.

3D Shapes Geometry

Delving into 3D shapes geometry allows students to explore the world of solids from cubes to complicated polyhedra. In 3D geometry, we recognize solids by identifying features like edges, vertices, and the angles between faces. Through this discipline, students learn how different shapes can be combined or transformed, like with truncation or stretching. These transformations alter the shape's geometry, changing its properties and appearance.

The truncated octahedron stands as a wonderful example of geometric transformation. Initially, an octahedron features eight triangular faces; after truncation, it becomes a new solid with six square and eight hexagonal faces. This exercise in geometry helps enhance spatial reasoning and understanding of how geometric operations modify structures. Envisioning how shapes morph in 3D space is a key skill in fields ranging from architecture to computer graphics.