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Chapter 12: Problem 7

Name the polygons that form the lateral faces of each polyhedron. rectangular pyramid

Short Answer

Expert verified
The lateral faces of a rectangular pyramid are all triangles.

Step by step solution

01

Understand the Rectangular Pyramid

A rectangular pyramid is a three-dimensional shape with a rectangular base and four triangular lateral faces that extend from each side of the base to meet at a single point, known as the apex.
02

Identify the Lateral Faces

The lateral faces are the sides of the pyramid that connect the edges of the base to the apex. In the case of a rectangular pyramid, these lateral faces are all triangles.
03

Count and Confirm the Triangular Faces

Since the base of the pyramid is a rectangle, there are four triangular faces in total, as each side of the rectangle forms the base of one triangular face.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Three-Dimensional Shapes
Three-dimensional shapes, also known as 3D shapes, are objects that have length, width, and depth; they occupy space and have volume. A rectangular pyramid is a classic example of a three-dimensional shape. It consists of a base and apex positioned above the base that connects with the four triangular lateral faces.
These shapes are intrinsic to geometry as they help describe and model various physical objects and spaces around us.
  • Length, Width, and Depth: Different from two-dimensional shapes, 3D shapes have three dimensions. For a rectangular pyramid, length and width define the rectangle base, while depth adds to its height from the base to the apex.
  • Volume and Surface Area: While 2D shapes have area, 3D shapes have both surface area and volume. For a rectangular pyramid, surface area includes all its triangular faces and its rectangular base, whereas volume measures the space within the pyramid.
Triangles
Triangles are a fundamental part of many geometric structures and are the simplest form of polygons, with three sides and three vertices. In a rectangular pyramid, the triangular faces form the lateral surfaces, one extending from each side of the base to meet at the apex.
These triangles transform a simple rectangle into a more complex three-dimensional form by bridging the base and the apex. Understanding triangles is key to grasping the concept of pyramids.
  • Labels and Terms: In a rectangle-based pyramid, triangles are identified by their base (one edge of the rectangle) and their apex, which forms the opposite vertex.
  • Types of Triangles: Generally, the triangles found in pyramids are of the scalene type because their sides can all be of different lengths unless specified otherwise.
Polyhedra
Polyhedra are three-dimensional shapes with flat polygonal faces, straight edges, and sharp vertices. They can have various forms, with the pyramid being one of the simplest types. A rectangular pyramid counts as a polyhedron due to its polygon-based structure comprising triangles and rectangles.
This class of shapes is particularly relevant in mathematics and engineering, serving as a building block for more complicated structures.
  • Faces, Edges, and Vertices: Polyhedra feature flat faces, where each intersection forms an edge. In a rectangular pyramid, there are five faces, eight edges, and five vertices, including the apex.
  • Classification: Polyhedra are classified by the shape of their faces. A rectangular pyramid is specifically a non-regular polyhedron because its faces are not all identical.

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