Introduction

Relevance of the Topic

Spatial Geometry is a bridge between plane figures and three-dimensional space, helping to understand how shapes can exist in different dimensions and providing a basis for the study of complex solids. The surface area of a prism is a fundamental concept, applying from civil construction, where regular prisms (cube, parallelepiped, etc.) are common volume units, to packaging design, where calculating the surface area of a prism is vital to determine the amount of material needed for its manufacture.

Contextualization

The surface area of a prism is a natural extension of calculating the area of plane figures. We approach the prism after plane figures, as the base of the prism is a plane figure, which is repeated along a dimension. Thus, the prism allows us to visualize and quantify how plane areas can accumulate in a three-dimensional space. This is crucial for 12th-grade students, as they are solidifying their geometry knowledge before moving on to more advanced topics, such as quadratic equations and trigonometry.

Theoretical Development

Components

• Prisms: Prisms are geometric solids formed by two parallel bases and flat lateral faces. A base is a plane figure that is repeated and translated to form the prism. The lateral faces are rectangles that "wrap" around the bases, and the faces of the prism are the rectangles and the bases. The surface area of a prism is the sum of the areas of all its faces.

  • Regular Prisms: These are prisms where all lateral faces are congruent rectangles. If the base of the prism is a regular polygon and the lateral edges are all the same length, the prism is regular.

• Area of Plane Figures: To calculate the surface area of a prism, we use prior knowledge of how to calculate the area of plane figures. In the case of a regular prism, the surface area is simply the area of the base multiplied by the number of lateral faces. For the plane figures that are bases of the prism, we use the corresponding area formulas.

Key Terms

• Area: In geometry, area is the measure of how large a surface is. It is the amount of space within closed boundaries. The unit of measure for area in the International System of Units (SI) is the square meter (m²).

• Base of the Prism: The base of a prism is the plane figure that is repeated and translated to form the prism. In a regular prism, the base is a regular polygon.

• Face of the Prism: A face of the prism is one of the planes that form the "walls" of the prism. In the case of a regular prism, the faces consist of rectangles (the lateral faces) and the regular polygon of the base.

• Volume of the Prism: Volume is the amount of space occupied by a three-dimensional object and is measured in cubic units. In a prism, the volume is calculated by multiplying the area of the base by the height of the prism.

Examples and Cases

• Cases of Regular Prisms:

  • Cube: A cube is a regular prism where all faces are congruent squares. The surface area of a cube is 6 times the square of the length of an edge.
  • Parallelepiped: A parallelepiped is a regular rectangular prism where all lateral faces are congruent rectangles. The surface area of a parallelepiped is the sum of twice the product of the height and width, twice the product of the height and length, and twice the product of the width and length.

• Application in Packaging: In packaging projects, calculating the surface area of a prism is vital to determine the amount of material needed for the manufacture of the packaging. For example, a cuboid box (a type of rectangular prism) needs a specific amount of cardboard to cover all its faces, and this calculation is made from the surface area of the prism.

• Use in Civil Construction: In civil construction, the surface area of the prism is an essential concept. For example, to cover the walls of a room with paint, it is necessary to know the total area of the walls, which is basically the surface area of the prism formed by the walls.

Detailed Summary

Relevant Points

• Importance of Prisms: Prisms are relevant and common geometric solids in various applications, from civil construction to packaging design. They are formed by two parallel bases and flat lateral faces and allow us to visualize how plane areas accumulate in a three-dimensional space.

• Area of Regular Prisms: For a regular prism, calculating the surface area is relatively straightforward and composed of the sum of the area of the base and the area of the lateral faces. The base is a regular polygon and the lateral faces are congruent rectangles.

• Calculation of the Base Area: Calculating the area of the base involves knowledge of the polygon that constitutes it. For squares, it is the square of the side, for rectangles it is the product of the base by the height, and so on.

• Calculation of the Lateral Faces Area: The area of the lateral faces in a regular prism is calculated by multiplying the perimeter of the base by the height of the prism.

Conclusions

• Versatility of Area Calculation: The calculation of the surface area of the prism shows the versatility and practical application of the concepts of calculating areas of plane figures. Understanding this operation allows for the accumulation of knowledge, being a fundamental step for future studies in geometry and mathematics in general.

• Importance of Spatial Geometry: The surface area of the prism is a crucial component of Spatial Geometry, which is situated in the midst of a wide range of applications and fundamental concepts, which have a direct impact on various areas, such as engineering, architecture, and physics.

Exercises

1. Calculating the Area of the Prism: Given a rectangular prism with a base of 4 meters by 5 meters and a height of 3 meters, calculate its surface area.

2. Identifying Regular Prisms: In a classroom, find 3 objects that are regular prisms. Describe the characteristics of each and cite the formula used to calculate the area of their surface.

3. Practical Application: If you had to decorate the walls of a room with wallpaper, how would you calculate the necessary amount, considering that the room is 3 meters high and measures 4 meters by 5 meters in the plan?