Problem 27 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

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

Step-by-Step Solution

Answer

The truncated octahedron has 8 hexagonal faces.

Step 1: Understanding the Truncated Octahedron

A truncated octahedron is a polyhedron that can be obtained by truncating (cutting off) the corners of an octahedron. It consists of faces that are either hexagonal or square.

Step 2: Components of a Truncated Octahedron

A truncated octahedron is made up of two different types of faces: hexagons and squares. We need to identify each type to determine how many of them there are.

Step 3: Identifying the Hexagonal Faces

For the truncated octahedron, each vertex of the original octahedron is truncated, creating a hexagonal face. Since an octahedron has 6 vertices and each is truncated to form a hexagonal face in the truncated form, the truncated octahedron contains 8 hexagonal faces.

Key Concepts

Understanding Polyhedrons

A polyhedron is a three-dimensional shape with flat polygonal faces, straight edges, and vertices. These fascinating structures can take on numerous forms and have a rich presence in both mathematics and nature.
Think of a polyhedron as a multi-sided die where each face is a polygon. Polyhedrons are often classified based on their face shapes, numbers of edges, and vertices.

• Convex Polyhedrons: All faces point outward, like cubes.
• Concave Polyhedrons: Have some inward-pointing faces.
• Regular Polyhedrons: Faces and angles are identical, like tetrahedrons.
• Truncated Polyhedrons: Made by cutting (truncating) parts off other polyhedrons.

Polyhedrons feature prominently in geometric studies and have practical applications in fields ranging from architecture to molecular biology.

The Role of Hexagonal Faces

Hexagonal faces are a crucial aspect of many polyhedrons, including the truncated octahedron. A hexagon is a six-sided polygon, and in three-dimensional shapes, these faces provide stability and aesthetic appeal.
In the context of a truncated octahedron, hexagonal faces emerge from the process of truncating an octahedron's vertices. Each vertex cut generates a hexagonal face, replacing the original triangular faces of the octahedron.

• Hexagons allow more efficient space packing, which is why honeycombs use them.
• They maintain equal angles and side lengths, providing uniformity.
• Truncated octahedrons specifically have 8 hexagonal faces, making them unique among polyhedrons.

Understanding hexagonal faces helps in visualizing and constructing complex geometrical models which have implications in various scientific and engineering disciplines.

Exploring the Octahedron

An octahedron is a type of polyhedron that has eight faces. It's a fascinating shape that can be visualized as two pyramids base-to-base.
This shape naturally features equilateral triangle faces, and it's known for its symmetry properties.

• An octahedron has 6 vertices.
• Its symmetrical nature allows equal division and even shaping.
• It can be truncated to form other polyhedrons, like the truncated octahedron.

The octahedron's ease of division and application in modeling molecular geometry make it a staple in both academic and practical applications. Learning how truncating these shapes alters their properties and structures is key to understanding more advanced geometric concepts.