Introduction

The Relevance of the Topic

The concept and application of volume and area of the cylinder are extremely relevant in Mathematics and in various other areas that depend on this discipline: Science, Engineering, Architecture, among others. They allow us to calculate the amount of three-dimensional space that a cylindrical object occupies (volume) and the two-dimensional extent of its lateral surface (area). Furthermore, this ability to visualize and understand complex spatial objects like the cylinder is a key skill to develop spatial and abstract reasoning.

Contextualization

Located in the unit of Spatial Geometry, the study of the volume and area of the cylinder is a natural continuation of previous learning about flat figures. Here, we expand the focus from two-dimensional calculations to three-dimensional calculations, deepening the perception of how solid figures relate and transform. By understanding the volume and area of the cylinder, you will be building a solid foundation to approach more complex concepts, such as the sphere and the cone, as well as expanding your abstraction and visualization in space.

This learning is fundamental for the development of mathematical and scientific education, empowering our students to solve real-world problems ranging from calculating the volume of a water reservoir to calculating the area of a soda can for the placement of a label - practical applications that, among others, are the result of understanding the volume and area of the cylinder.

Theoretical Development

Components

• What is a Cylinder?

  • A cylinder is a spatial geometric figure that has two congruent parallel bases and a curved surface that connects them. The base and the top of the cylinder are circles, and the curved line that connects them is called the lateral surface. This is fundamental for us to understand the structure and properties of the cylinder.

• Volume of the Cylinder

  • The volume of a figure is the amount of space contained within it. For the cylinder, the volume is calculated by multiplying the area of the base (since the bases of the cylinder are circles) by the height of the cylinder. That is, V = A_base × h, where A_base = πr² and r is the radius of the base.
  • The formula for the volume of the cylinder is commonly remembered by the expression "Pir^2h", where PI (π) represents the approximate value 3.1416 and r is the radius of the base, and h is the height of the cylinder.

• Area of the Cylinder

  • The area of a figure is the measure of the surface covered by it. In the case of the cylinder, the area is the sum of the areas of the lateral surface (which is a rectangle with a height equal to the height of the cylinder and a length equal to the circumference of the base) with the areas of the two bases (circles). The formula to calculate this sum is: A = 2πrh + 2πr², where r is the radius of the base, h is the height of the cylinder, and π is approximately 3.1416.
  • It is worth noting that the formula for the area of the cylinder is a combination of the formulas for the area of the circle (πr²) and the rectangle (2h) - elements that manifest in the geometry of the cylinder.

Key Terms

• Radius (r):

  • The radius is the distance from the center of the circle to any point on its circumference. In the cylinder, the radius is the same both at the base and at the top.

• Height (h):

  • Height is the distance between the two bases. In the cylinder, the height is perpendicular to the base and the top and, when extended, can be seen as the height of a rectangle that surrounds the lateral surface.

• Lateral Surface:

  • Surface that connects the two bases of the cylinder. This surface is curved and symmetrical, and its shape is a rectangle when extended.

• Bases (or Base Circles):

  • The two congruent flat faces that define the tops and the base of the cylinder. In the cylinder, the bases are circles.

Examples and Cases

• Example of Calculating the Volume of a Cylinder:

  • Imagine a cylinder with a radius of 4cm and a height of 6cm. Applying the formula V = πr²h, we have: Volume = 3.1416 x 4² x 6 = 301.59cm³.
  • Here, you are measuring the amount of 3D space (cm³) that this cylinder occupies.

• Example of Calculating the Area of a Cylinder:

  • For the same cylinder above, the formula A = 2πrh + 2πr² gives us: Area = 2 x 3.1416 x 4 x 6 + 2 x 3.1416 x 4² = 225.6cm².
  • In this case, you are measuring the two-dimensional extent (cm²) of this figure.

• Practical Application: Amount of Paint to Paint a Cylinder:

  • If you need to paint the cylinder from the previous example with paint that covers 10m² per liter and the cylinder is made of material that requires two coats of paint, then how many liters of paint will you need? Apply the area calculation to find out!
  • Total area = Lateral surface area + Bases area = 225.6cm² + 2 x 50.24cm² (we use the circle area formula to calculate) = 326.08cm² = 0.032608m².
  • As two layers of paint are required, the area is doubled: 0.032608m² x 2 = 0.065216m².
  • Therefore, the amount of paint needed will be: 0.065216m² / 10m²/l = 0.0065216 l = 652.16 ml.

• Unfolding: Cylinders and Cones in Real Life:

  • Cylinders are everywhere in everyday life - soda cans, cups, pipes - and the knowledge about the area and volume of a cylinder can be extended to calculate these measures in other cylindrical objects.
  • Furthermore, the theoretical results about the volume and area of the cylinder can be used to understand other solids of revolution, such as the cone, where the shape is a rotation of a right triangle around an axis.

Detailed Summary

Key Points

• Definition of Cylinder:

  • The cylinder is a spatial geometric figure with two congruent parallel bases, which are circles, and a curved surface that connects them. The structural properties of the cylinder provide the basis for understanding the calculation of volume and area.

• Volume of the Cylinder:

  • The volume of a cylinder is calculated by multiplying the area of the base by the height: V = A_base × h. The base of the cylinder is a circle, so the volume formula is πr²h, where r is the radius of the base and h is the height of the cylinder.
  • The volume is calculated in cubic units, reflecting the three-dimensional space that the object occupies.

• Area of the Cylinder:

  • The area of the cylinder is the sum of the area of the two bases (two circles) with the area of the lateral surface (a rectangle that extends around the cylinder). The formula to calculate the area is A = 2πrh + 2πr², where r is the radius of the base and h is the height of the cylinder.
  • The area is calculated in square units, indicating the two-dimensional extent of the object's surface.

• Practical Applications:

  • Knowledge about the volume and area of the cylinder has practical applications in various everyday and real-world situations. For example, we need these measures to determine the amount of paint to paint a soda can or to calculate the capacity of a cylindrical reservoir.

Conclusions

• General Skills:

  • The study of the volume and area of the cylinder is an opportunity to develop general skills in Mathematics, such as spatial visualization, logical reasoning, and problem-solving. These skills are transferable to many other topics and disciplines.

• Important Formulas:

  • The volume of the cylinder and the area formulas, V = πr²h and A = 2πrh + 2πr², respectively, are key concepts that must be mastered. We must remember that the radius (r) is the distance from the center of the circle to any point on its circumference and the height (h) is the distance between the two bases of the cylinder.

Suggested Exercises

1. Calculating the Volume:

   • Given a cylinder with a radius of 2cm and a height of 10cm, calculate its volume.
   • Tip: Remember that the formula to calculate the volume of the cylinder is V = πr²h.

2. Calculating the Area:

   • For the same cylinder from the previous exercise, what is the total area?
   • Tip: Remember that the formula to calculate the area of the cylinder is A = 2πrh + 2πr².

3. Practical Application:

   • If you have a standard soda can, with a radius of 3.2 cm and a height of 12.2 cm, and want to know how many milliliters of soda it contains, assuming it is completely full, what calculations would you use to get the answer?
   • Tip: Use the formula for the volume of the cylinder and remember that 1 ml = 1 cm³.