Surface Area Ratio of a Right Circular Cylinder: Exploring the Relationship Between Total and Curved Surface Areas

Introduction

In this article, we delve into the fascinating world of geometric objects, specifically focusing on the properties of a right circular cylinder. This shape is common in various fields, including architecture, engineering, and mathematics. We will explore the concepts of the total surface area and the curved surface area of a cylinder and examine the ratio between them. Specifically, we will calculate the ratio for a cylinder with a base radius of 5.7 cm and a height of 9.7 cm.

Understanding the Cylinder

A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. The key elements of a cylinder include its base radius (r), the distance from the center of the base to the edge, and height (h), the distance between the two bases. The problem presented here involves a cylinder with a base radius of 5.7 cm and a height of 9.7 cm.

Total Surface Area and Curved Surface Area

The total surface area (TSA) of a cylinder is the sum of the areas of the two circular bases and the curved surface. Mathematically, it is given by the formula:

TSA 2πr(r h)

The curved surface area (CSA), on the other hand, is the area of the curved surface connecting the two bases. It is given by the formula:

CSA 2πrh

Both of these formulas are derived from the basic area and circumference formulas of a circle. The perimeter of the circular base (2πr) is multiplied by the height (h) to find the CSA, and twice that perimeter, along with the area of the two bases, gives the TSA.

Calculating the Ratio

To find the ratio of the total surface area to the curved surface area, we start with the given formulas:

TSA 2πr(r h)

CSA 2πrh

The ratio of the total surface area to the curved surface area is then calculated as:

TSA : CSA [2πr(r h)] : [2πrh]

By simplifying the above expression, we get:

TSA : CSA [r h] : h

Substituting the values r 5.7 cm and h 9.7 cm, we get:

TSA : CSA [5.7 9.7] : 9.7

Thus, the ratio becomes:

TSA : CSA 15.4 : 9.7

This can be simplified further by dividing both terms by 9.7:

15.4 / 9.7 1.59

Therefore, the final ratio is approximately 1.59.

The ratio can also be expressed in a more traditional mathematical form as:

TSA : CSA 24 : 13

This simplification occurs because both terms in the ratio can be multiplied by 1.59, and this multiplication does not change the nature of the ratio.

Conclusion

Understanding the relationship between the total and curved surface areas of a cylinder is crucial for many applications in engineering, architecture, and even everyday life. By using the given formulas and performing the necessary calculations, we can easily determine the ratio for any given cylinder's dimensions. The specific example we solved here, with a base radius of 5.7 cm and a height of 9.7 cm, demonstrates a clear and concise method for finding the desired ratio.

For those interested in further exploration, you might consider experimenting with different values for the radius and height to see how the ratio changes, or even working on more complex shapes that incorporate multiple cylinders.