Question

Lateral surface area of prisms. Theorem. The lateral surface area of a prism is equal to the product of a lateral edge and the perimeter of a perpendicular cross section. By a perpendicular cross section (Figure 51) of a prism, we mean the polygon abcde obtained by intersecting all lateral faces of the prism by a plane perpendicular to the lateral edges. Sides of this polygon are perpendicular to the lateral edges \((\S \S 31,20)\). The lateral surface area of the prism is equal to the sum of areas of parallelograms. In each of them, a lateral edge can be considered as the base, and one of the sides of the perpendicular cross section as the altitude. Therefore the lateral surface area is equal to \(A A^{\prime} \cdot a b+\) \(B B^{\prime}+b c+C C^{\prime} \cdot c d+D D^{\prime} \cdot d e+E E^{\prime} \cdot e a=A A^{\prime} \cdot(a b+b c+c d+d e+e a)\). Corollary. The lateral surface area of a right prism is equal to the product of the perimeter of the base and the altitude, because lateral edges of such a prism are congruent to the altitude, and its base can be considered as the perpendicular cross section.

Short answer

The lateral surface area is found by multiplying the lateral edge by the perimeter of a cross-sectional polygon.

Step-by-step solution

Understand the Problem

The lateral surface area of a prism is determined by the formula which multiplies the lateral edge by the perimeter of a perpendicular cross section.

Identify the Components

Recognize the components of the formula: the lateral edge, and the perimeter of a cross-sectional polygon that is perpendicular to the lateral edge.

Formula Application

Given: Lateral edge = AA', and the sides of the cross-section = ab, bc, cd, de, ea. The formula for lateral surface area is: \[ A = A A' \cdot (ab + bc + cd + de + ea) \].

Understanding the Corollary

For a right prism, the lateral surface area simplifies because the perpendicular cross-section is the base, and the lateral edge equals the prism's height. Hence, the lateral surface area becomes the product of the base's perimeter and the prism's height.

Key concepts

Perpendicular Cross Section

A perpendicular cross section in the context of prism geometry is a crucial concept. Imagine slicing a prism with a plane so that it is exactly at a 90-degree angle to the lateral edges of the prism. This slice creates a flat shape, which is the cross-section, and is often a polygon such as a triangle or rectangle.

This cross section is vital because:

• It helps in calculating the lateral surface area.
• The sides of the cross-section are perpendicular to the prism’s lateral edges.
• The perimeter of this shape is used in the lateral surface area formula.

Understanding the perpendicular cross section allows us to more easily compute other aspects of the prism, such as the surface area and volume.

Prism Geometry

Prism geometry is all about understanding the 3-dimensional shape called a prism. At its core, a prism is a solid object that has two identical flat ends (called bases) and faces (the sides that connect the bases) that are parallelograms.

Key characteristics of prism geometry include:

• Base Shape: This determines the name of the prism, such as a triangular prism or a rectangular prism.
• Lateral Edges: These are the edges connecting the corresponding vertices of the bases.
• Uniform Cross-Section: The cross-sectional area remains the same along the length of the prism.

Studying the geometry of prisms not only helps us with area calculations but also with understanding the spatial properties that define different types of prisms.

Right Prism Properties

A right prism is a special type of prism where the lateral edges are perpendicular to the bases. Hence, the cross-sections made perpendicular to the lateral edges are identical to the base.

Right prism properties that are important:

• Altitude Equals Lateral Edge: The height of the prism is the same as the length of its lateral edges.
• Base and Cross Section: For a right prism, the base serves as the perpendicular cross section.
• Lateral Surface Area Calculation: For a right prism, the lateral surface area is straightforward. It is the product of the base perimeter and the height of the prism.

Analyzing right prism properties simplifies the computation of surface areas, making it easier for students to tackle geometry problems involving these shapes.