Introduction

Relevance of the Topic

The Volume of the Cylinder is a crucial part of Spatial Geometry and has immense practical application. From calculating the volumes of containers in industries, such as soda or oil cans, to measuring the volume of grains stored in silos, these are examples of situations where understanding and applying this concept become indispensable. Moreover, the study of the cylinder is directly connected to other important solids in geometry, such as the cone and the sphere, thus becoming a foundation for more advanced knowledge.

Contextualization

Within Mathematics, the topic is situated within Spatial Geometry, which in turn is one of the most applied and practical areas of the discipline. This topic is the natural continuation of the study of the Volume of the Parallelepiped and the Prism, deepening the concept of the relationship between the measurements of a solid and its volume. In the curriculum, the study of the Volume of the Cylinder comes right after the understanding of similar solids and before complications such as "solids of revolution", which include the cone and the sphere - topics that heavily depend on the understanding of the cylinder.

Theoretical Development

Components

• Cylinder: A cylinder is a solid of revolution formed by all points at a fixed distance from a coplanar line segment to it, this segment is the generatrix, and a plane perpendicular to it and containing it, these planes are the bases of the cylinder. The cylinder is a solid that arises in practice in various situations: from pipes and tubes to soda cans.

• Base of the Cylinder: The bases of a cylinder are two identical circles, one at the beginning and the other at the end of the line segment (generatrix) that is perpendicular to them. The base is a fundamental element for calculating the area and volume of the cylinder.

• Height of the Cylinder: The height of a cylinder is the distance between its bases and does not necessarily coincide with the generatrix. It is a fundamental value for calculating the volume of the cylinder.

• Generatrix of the Cylinder: The generatrix of a cylinder is the line segment that connects the center of one base to the center of the other. It is an element that helps characterize the cylinder, but does not directly interfere in the calculation of the volume.

• Radii of the Cylinder Bases: The radius of a circle is the distance from the center of the circle to any point on its perimeter. In a cylinder, the radius of the bases is the same, and this value is fundamental for calculating the area and volume.

Key Terms

• Volume: In geometry, volume is the amount of three-dimensional space occupied by a solid. In the case of the cylinder, the formula for calculating the volume is the area of the base multiplied by the height: V = A * h.

• Pi (π): Pi is a mathematical constant that represents the ratio between the circumference of any circle and its diameter. It is an irrational number, which means it never ends or repeats its exact decimal sequence.

• Area of the Circle: The area of a circle is calculated by multiplying the radius squared by the constant pi: A = π * r². In the case of the cylinder, as the bases are circles, the area of the base used in the calculation of the volume is π * r².

Examples and Cases

• Case 1: Imagine a cylinder with a base radius measuring 3 meters and a height of 5 meters. To calculate the volume of this cylinder, we use the formula V = π * r² * h, where r = 3 and h = 5. Therefore, the volume of this cylinder is V = 3.14 * 3² * 5 = 141.3 cubic meters.

• Case 2: Suppose an oil can whose height is 35cm and the base radius is 5cm. To find the volume of this can, just apply the formula V = π * r² * h, considering r = 5cm and h = 35cm. Therefore, the volume of the oil can is V = 3.14 * 5² * 35 = 2747.5 cm³.

In these examples, we can clearly see the application of the cylinder volume formula, the use of the concept of pi, and the crucial role of the radius and height measurements in determining the volume.

Detailed Summary

Relevant Points:

• Structure of the Cylinder: A cylinder is a solid of revolution formed by two parallel circular bases and a cylindrical envelope, which is perpendicular to the bases. The perimeter of the base and the height of the cylinder are the main factors to be considered in calculating its volume.

• Cylinder volume formula: The formulation for calculating the volume of a cylinder is V = π * r² * h, where V represents the volume, π the constant pi (approximately 3.14), r is the base radius and h is the height of the cylinder.

• Relationship between Cylinders and Circles: Circles are important in the study of the volume of the cylinder because the bases of a cylinder are circles. The area of a circle (A = π * r²) is used for the calculation of the cylinder's volume, multiplying it by the height.

• Cases and Applications: The volume of the cylinder is a versatile and applicable concept in many everyday situations, from calculating the volume of cans and tubes to measuring quantities of grains in silos.

Conclusions:

• Inseparability of the Cylinder and the Circle: The cylinder is essentially a "stretched" circle on the third axis. This relationship, coupled with the constant pi, allows for an easy transposition from the study of the circle to the understanding and calculation of the cylinder's volume.

• Importance of Practice: The ability to calculate the volume of the cylinder is a key skill that can only be improved with practice. Solving varied exercises in different contexts is essential for consolidating this concept.

• Solids of Revolution: The study of the cylinder's volume is a natural precursor to understanding and calculating the volumes of other solids of revolution, such as the cone and the sphere. Mastery of this topic is, therefore, a critical component of a solid foundation in spatial geometry.

Exercises:

1. Exercise 1: Calculate the volume of a cylinder with a base radius measuring 2 meters and a height of 6 meters.

2. Exercise 2: A paint can has a height of 15 centimeters and the base radius measures 4 centimeters. What is the volume of this paint can?

3. Exercise 3: Find the height of a cylinder with a volume of 2000 cubic meters and a base radius measuring 10 meters.