Volume: Contextualized Rectangular Prism | Traditional Book

Prisms are geometric shapes that we often encounter in our daily lives, even if we don't realize it most of the time. For example, the shoebox you use to store your sneakers, the water tank in your backyard, and even the structure of a building are examples of prisms. Calculating the volume of these structures is essential in various fields, such as civil engineering and packaging design, as it helps determine the amount of material needed and storage capacity.

Think About: Have you ever stopped to think about how engineers calculate the amount of concrete needed to build a building or how manufacturers know exactly how much product fits inside a package?

Geometry is a branch of mathematics that studies shapes, sizes, and properties of spaces. Within this area, the study of geometric solids, such as prisms, is fundamental for various practical applications. A prism is a three-dimensional solid with two parallel and congruent bases, and lateral faces that are parallelograms. Rectangular prisms, also known as rectangular parallelepipeds, have rectangular bases and are common in everyday objects like boxes and containers.

Calculating the volume of a prism is an important skill that allows you to determine the storage capacity of an object or the amount of material needed to fill it. In the case of rectangular prisms, the volume is obtained by multiplying the length, width, and height of the prism. For triangular-based prisms, the volume is calculated based on the area of the triangular base multiplied by the height of the prism.

Understanding how to calculate the volume of prisms is essential not only for solving academic problems but also for facing practical challenges in everyday life. For instance, when planning to construct a cistern to store rainwater, it is necessary to know the volume of the cistern to ensure it has sufficient capacity. Similarly, when developing packaging for products, it is crucial to calculate the volume to optimize space usage and reduce costs. In this chapter, we will deepen our understanding of calculating the volume of prisms, exploring practical examples and contextualized problems to consolidate this understanding.

Definition of Rectangular Prisms

Rectangular prisms, also known as rectangular parallelepipeds, are geometric solids with two parallel and congruent rectangular bases, and four lateral faces that are rectangles. Each edge of the bases is connected by a lateral edge, forming a right angle with the base. Therefore, all lateral faces are perpendicular to the bases. This structure makes rectangular prisms quite common in everyday objects, like shoeboxes, drawers, and containers.

To better understand a rectangular prism, imagine a shoebox. The base of the box can be considered a rectangle, and the lid of the box is another identical rectangle, positioned parallel to it. The lateral faces of the box are formed by the walls of the box, which are also rectangles. This configuration characterizes a rectangular prism. Mathematically, a rectangular prism has six faces, twelve edges, and eight vertices.

In addition to the structural definition, it is important to understand that rectangular prisms are three-dimensional solids, meaning they have volume. The volume of a rectangular prism is a measure of how much space it occupies in three-dimensional space. In the case of rectangular prisms, calculating the volume is straightforward, using the formula V = length x width x height. This formula is derived from multiplying the three dimensions of the rectangle that form the base of the prism and its height.

Formula for the Volume of Rectangular Prisms

The formula for calculating the volume of a rectangular prism is V = length x width x height, where V represents the volume, and length, width, and height are the dimensions of the prism. This formula is intuitive because it multiplies the three dimensions of the prism to determine the three-dimensional space it occupies. The length and width define the area of the rectangular base, while the height indicates how many of these bases fit in the prism.

To illustrate the application of this formula, consider a rectangular prism with a length of 8 cm, a width of 5 cm, and a height of 10 cm. Applying the formula, we have V = 8 cm x 5 cm x 10 cm, resulting in a volume of 400 cm³. This means the prism occupies a space of 400 cubic centimeters. This type of calculation is useful in various practical situations, such as determining the capacity of a box to store objects or calculating the amount of material needed to fill a container.

Another practical application of calculating the volume of rectangular prisms is in civil engineering. For example, when planning the construction of a concrete foundation, it is necessary to calculate the volume of the area to be filled. If the foundation is in the shape of a rectangular prism, the volume can be quickly calculated using the formula. This helps estimate the amount of concrete needed, avoiding waste and ensuring the project's efficiency. Thus, the volume formula for rectangular prisms is an essential tool for both students and professionals.

Triangular-Based Prisms

Triangular-based prisms are geometric solids that have two congruent and parallel triangular bases, connected by three lateral faces that are rectangles. Just like rectangular prisms, triangular-based prisms are three-dimensional solids but differ in the shape of their bases. Each edge of the triangular bases is connected by a lateral edge, forming a solid with five faces, nine edges, and six vertices.

To better understand a triangular-based prism, imagine a Toblerone, the well-known chocolate bar. The base of each triangular section of the Toblerone is a triangle, and the lateral faces are rectangles connecting these triangles. This configuration characterizes a triangular-based prism. Mathematically, the added complexity compared to rectangular prisms is that the base is not a rectangle but a triangle, which requires a different calculation to determine the volume.

Calculating the volume of a triangular-based prism involves the area of the triangular base. The formula for the volume is V = base area x height, where the height is the perpendicular distance between the two triangular bases. This formula is derived from the principle that the volume of the prism is the base area multiplied by the height, similar to rectangular prisms but adjusted for the triangular geometry of the base. This formula is crucial for solving problems involving triangular-based prisms in practical and academic contexts.

Practical Examples of Volume Calculation

To consolidate the understanding of calculating the volume of prisms, let's explore some practical examples. Consider a shoebox with dimensions of 30 cm in length, 20 cm in width, and 15 cm in height. Applying the volume formula for rectangular prisms, we have V = length x width x height. Substituting the values, V = 30 cm x 20 cm x 15 cm, which results in a volume of 9000 cm³. This means the shoebox can hold 9000 cubic centimeters of space, which can be used to store shoes or other items.

Another practical example is calculating the volume of a water tank to store rainwater. Suppose the water tank is in the shape of a rectangular prism with dimensions of 2 meters in length, 1.5 meters in width, and 1 meter in height. Applying the formula, we have V = length x width x height, substituting the values, V = 2 m x 1.5 m x 1 m, resulting in a volume of 3 cubic meters. Since 1 cubic meter equals 1000 liters, the volume of the water tank is 3000 liters. This calculation is essential to ensure that the water tank has sufficient capacity to store the desired amount of water.

In the case of triangular-based prisms, consider a prism made of cardboard with a triangular base whose area is 12 cm² and a height of 7 cm. The formula for the volume is V = base area x height. Substituting the values, V = 12 cm² x 7 cm, resulting in a volume of 84 cm³. This calculation is useful in situations like packaging production or constructing geometric models in school projects. With these practical examples, it becomes clear how the theoretical concepts of the volume of prisms are applied in various everyday situations.

Reflect and Respond

• Think about how the calculation of the volume of prisms can be applied in different professions, such as civil engineering, pharmacy, and packaging design.
• Reflect on the importance of understanding the concept of volume and its practical application in daily life, such as in planning storage and constructing structures.
• Consider how understanding the volume of prisms can help solve real problems and make informed decisions in practical situations, such as calculating the capacity of a water tank or packaging.

Assessing Your Understanding

• Explain how to calculate the volume of a rectangular prism and a triangular-based prism, providing practical and contextualized examples.
• Describe a situation in your daily life where you needed to calculate the volume of an object. How did you apply the volume formulas for rectangular prisms or triangular-based prisms to solve the problem?
• Discuss the importance of calculating volume in civil engineering and other professional fields. How can the knowledge acquired in this chapter be applied in various contexts?
• Compare the differences between rectangular prisms and triangular-based prisms. What are the main distinctions in their structures and volume calculations?
• Analyze a contextualized problem that involves calculating the volume of a water tank or cistern. How would you approach solving this problem using the concepts learned in this chapter?

Reflection and Final Thought

Throughout this chapter, we explored in detail the calculation of the volume of rectangular prisms and triangular-based prisms, understanding their practical applications in various everyday contexts. The definition of rectangular prisms and triangular-based prisms allowed us to understand the structure of these geometric shapes and how they are found in common objects like shoeboxes and Toblerones. The application of the volume formulas, V = length x width x height for rectangular prisms and V = base area x height for triangular-based prisms, was illustrated with practical examples demonstrating the relevance of these calculations in real situations, such as in civil engineering and packaging design.

Understanding how to calculate the volume of these prisms is a crucial skill not only for solving academic problems but also for facing practical challenges in everyday life. Throughout the proposed activities, we reflected on the importance of the concept of volume and how it can be applied in different professions and contexts, from civil engineering to packaging production. This demonstrates the interconnection between mathematics and the real world, highlighting the utility of the knowledge acquired.

We encourage you to continue deepening your studies on calculating the volume of geometric solids, exploring other shapes beyond prisms and their practical applications. Mathematics is a science that helps us better understand the world around us and solve problems efficiently and informed. Keep practicing and applying these concepts to strengthen your understanding and mathematical skills.