Exploring the Diameter of a Cylinder with Equal Volume and Curved Surface Area

The problem of finding a cylinder where the volume is numerically equal to the curved surface area is an intriguing geometrical challenge. This article explores the calculation and the underlying principles to provide a clear understanding for students and professionals alike. Let's delve into the steps to solve this problem.

Given Conditions of the Cylinder

Let's consider a cylinder with a height of h 5 cm. Our goal is to determine its diameter d such that the numerical value of the volume is equal to the numerical value of the curved surface area.

Formulas and Equations

The volume of a cylinder is given by the formula:

V π r2 h

The curved surface area (CSA) of a cylinder is given by the formula:

CSA 2 π r h

Since the volume equals the curved surface area, we set the two formulas equal to each other:

π r2 h 2 π r h

Solving for the Diameter

By simplifying the equation, we can isolate h and π, which cancel out from both sides. This simplification leads to the following:

r2 2 r

Dividing both sides by r, where r is not zero, we get:

r 2

Since the diameter d is twice the radius, we have:

d 2 r 4 cm

This result is consistent regardless of the numerical value of the height h. Let's validate this by substituting a different height:

h 10 cm

V π r2 10

CSA 2 π r 10

Setting the volume equal to the curved surface area:

π r2 10 2 π r 10

Canceling out common factors (π, 10, r), we again find:

r 2

d 2 r 4 cm

This consistency confirms that the diameter of the cylinder must be 4 cm.

Conclusion

In conclusion, the diameter of a cylinder where the volume equals the curved surface area is consistently 4 cm, regardless of the height of the cylinder. This solution is derived from simple algebraic manipulations and the formulas for volume and surface area. Understanding this relationship not only enhances geometric insight but also provides a practical tool for solving similar problems in various applications.