Question

Name the polygons that form the lateral faces of each polyhedron. pentagonal prism

Short answer

The polygons are rectangles.

Step-by-step solution

Understand the Structure of a Pentagonal Prism

A pentagonal prism is a 3D shape with two parallel bases that are pentagons (five-sided polygons) and rectangular lateral faces that connect corresponding sides of the two bases.

Identify the Lateral Faces

Since the bases of a pentagonal prism are pentagons, there are five sides to each base. The lateral faces are the surfaces connecting the corresponding edges of these bases. Each lateral face is a rectangle.

Count the Number of Lateral Faces

Because a pentagonal prism has five sides on its base, there are five lateral faces, one for each pair of corresponding sides on the two parallel pentagonal bases.

Name the Polygons Forming the Lateral Faces

Since each of the five lateral faces is a rectangle, the polygons forming the lateral faces are rectangles.

Key concepts

Polygons

Polygons are fundamental shapes in geometry, consisting of straight lines that connect to form a closed loop. Each polygon can have three or more sides. Here are a few important characteristics of polygons:

• A polygon is named based on the number of its sides, such as triangles (3 sides), quadrilaterals (4 sides), and pentagons (5 sides).
• Regular polygons have all equal sides and equal angles, while irregular ones have sides and angles of different lengths and measures.
• Polygons primarily exist in a 2D plane, but they can be the faces of 3D shapes such as prisms and pyramids.

In the case of a pentagonal prism, which is the focus here, each base of the 3D shape is a polygon, specifically, a pentagon. This means each base has five straight sides, showcasing the simplest example of a polygon in three-dimensional geometry.

Prisms

Prisms are intriguing 3D shapes in geometry that are built from polygons. Each prism is defined by its two parallel bases which are congruent polygons. Here are key points about prisms:

• Prisms are named after the shape of their bases. For example, if the base is a pentagon, then it is called a pentagonal prism, like the one in our original exercise.
• The lateral faces of a prism are always parallelograms. For a right prism, which is most common, these are rectangles.
• In any prism, the number of lateral faces matches the number of sides on the base polygon. Thus, a pentagonal prism has five lateral faces, all of which are rectangles in a right prism arrangement.

Understanding prisms is crucial as they demonstrate how 2D shapes transition into 3D forms, preserving some basic properties of their base polygons.

3D Shapes

Three-dimensional shapes, or 3D shapes, exist in a space that gives them volume. Unlike 2D shapes that rest flatly on a plane, 3D shapes have thickness, length, and width. Important characteristics of 3D shapes include:

• The surfaces or faces of 3D shapes are usually polygons. For example, a cube has six square faces, while in our pentagonal prism, the faces are pentagons and rectangles.
• Edges are the line segments where two faces meet, and vertices are points where edges converge.
• Volume is another key aspect of 3D shapes, determining the capacity they hold, which is especially important in fields like architecture and physics.

For our pentagonal prism, the distinct features include five rectangular lateral faces and two pentagonal bases, illustrating how polygons are integral to the formation and understanding of more complex 3D shapes.