If A Cylinder Has A Height Equal To The Length Of Its Base Circumference, And The Height Is 125.66cm, What Are Its Radius, Diameter, And Volume?
Introduction
In the realm of geometry, the cylinder stands out as a fundamental three-dimensional shape. Its properties and characteristics are not only fascinating but also crucial in various real-world applications, from engineering and architecture to everyday objects we encounter. Understanding the relationship between a cylinder's dimensions, such as its height and the circumference of its base, is key to unlocking a deeper understanding of its nature. This article delves into a specific scenario: when a cylinder's height is equal to the circumference of its base. We will explore the implications of this condition, particularly when the height is given as 125.66 cm. Our exploration will cover the calculation of the cylinder's radius, diameter, and volume, providing a comprehensive analysis of its geometric properties.
The interplay between height and base circumference in a cylinder dictates its overall shape and volume. When these two measurements are directly linked, as in our case, it presents a unique geometric puzzle. The height, which is the perpendicular distance between the two circular bases, essentially defines the cylinder's vertical extent. The base circumference, on the other hand, is the distance around the circular base, determining its width. When these two dimensions are equal, it implies a specific proportion between the cylinder's height and its radius, leading to interesting mathematical relationships. In the following sections, we will unravel these relationships, employing fundamental formulas and principles of geometry to dissect this intriguing scenario. By the end of this article, you will gain a solid grasp of how to calculate key parameters of a cylinder when its height is equivalent to its base circumference, enhancing your understanding of three-dimensional geometry and its practical applications.
Understanding Cylinder Geometry
Before we dive into the specifics of our problem, let's first solidify our understanding of the basic geometry of a cylinder. A cylinder, at its core, is a three-dimensional shape with two parallel circular bases connected by a curved surface. Imagine a stack of identical circles placed one on top of the other – that's essentially what a cylinder is. The key parameters that define a cylinder are its radius (r), which is the distance from the center of the circular base to any point on its circumference, and its height (h), which, as we've mentioned, is the perpendicular distance between the two bases. From these two parameters, we can derive other important properties, such as the diameter (d), which is twice the radius (d = 2r), the circumference (C) of the base, given by the formula C = 2πr, and the area of the base (A), calculated as A = πr². These fundamental formulas are the building blocks for understanding and analyzing cylindrical shapes.
The relationship between these parameters is what defines the cylinder's overall form and size. The radius determines the width of the cylinder, while the height dictates its vertical extent. The circumference, being the distance around the circular base, is directly proportional to the radius. The area of the base, a measure of the space enclosed by the circular base, is also dependent on the radius. These relationships are not just abstract mathematical concepts; they have tangible implications in real-world scenarios. For instance, in engineering, the dimensions of a cylindrical pipe or tank directly influence its capacity and structural integrity. In architecture, cylindrical columns or towers need to be precisely designed to ensure stability and aesthetic appeal. Therefore, a solid understanding of cylinder geometry is essential for anyone working with three-dimensional shapes in practical applications.
Calculating the Radius
Now, let's tackle the core of our problem: determining the radius of the cylinder when its height is equal to the circumference of its base. We are given that the height (h) is 125.66 cm. According to the condition, h = C, where C is the circumference. We also know that the circumference is related to the radius (r) by the formula C = 2πr. By equating the height to the circumference, we get the equation 125.66 = 2πr. This equation is the key to unlocking the radius of our cylinder. Solving for r involves a simple algebraic manipulation: dividing both sides of the equation by 2π. Thus, r = 125.66 / (2π). To get a numerical value for the radius, we need to substitute the value of π, which is approximately 3.14159. Therefore, r ≈ 125.66 / (2 * 3.14159) ≈ 19.999 cm. For practical purposes, we can round this value to approximately 20 cm.
This calculation demonstrates a fundamental principle in geometry: using known relationships and equations to solve for unknown quantities. In this case, we leveraged the relationship between the circumference and the radius of a circle, along with the given condition that the height equals the circumference, to find the radius. The result, approximately 20 cm, provides a crucial piece of information about our cylinder. Knowing the radius allows us to calculate other important parameters, such as the diameter and the area of the base. Moreover, it gives us a visual sense of the cylinder's proportions – a radius of 20 cm coupled with a height of 125.66 cm suggests a relatively tall and slender cylinder. In the subsequent sections, we will build upon this knowledge to further explore the cylinder's properties and calculate its volume.
Determining the Diameter
With the radius now known to be approximately 20 cm, calculating the diameter is a straightforward process. The diameter of a circle, as mentioned earlier, is simply twice its radius. This relationship is a fundamental geometric principle, applicable to all circles regardless of their size. In mathematical terms, the diameter (d) is given by the formula d = 2r. Applying this formula to our cylinder, where r ≈ 20 cm, we get d = 2 * 20 cm = 40 cm. Thus, the diameter of the base of our cylinder is approximately 40 cm. This measurement represents the distance across the circular base, passing through its center. The diameter, along with the radius, provides a complete understanding of the size and proportions of the circular base.
The diameter is not just a numerical value; it has practical significance in various applications. For instance, in engineering, the diameter of a cylindrical pipe or shaft is a critical parameter in determining its strength and load-bearing capacity. In manufacturing, the diameter of a cylindrical component needs to be precisely controlled to ensure proper fit and function. In our specific case, knowing the diameter of 40 cm gives us a clear sense of the cylinder's width – it's a substantial dimension that contributes to the overall volume of the cylinder. Furthermore, the diameter, along with the height, can be used to visualize the cylinder's shape and proportions. A diameter of 40 cm and a height of 125.66 cm paint a picture of a cylinder that is significantly taller than it is wide, a characteristic that will influence its volume and other properties. In the next section, we will explore how to calculate the cylinder's volume, utilizing the radius and height that we have already determined.
Calculating the Volume
The volume of a cylinder is a measure of the three-dimensional space it occupies. It tells us how much a cylinder can hold, whether it's liquid, gas, or any other substance. The formula for the volume (V) of a cylinder is given by V = πr²h, where r is the radius of the base and h is the height of the cylinder. This formula is derived from the basic principle that the volume of a prism (and a cylinder is essentially a circular prism) is equal to the area of its base multiplied by its height. In our case, we have already determined that the radius (r) is approximately 20 cm and the height (h) is 125.66 cm. Substituting these values into the formula, we get V = π * (20 cm)² * 125.66 cm. Now, let's perform the calculation. First, we square the radius: (20 cm)² = 400 cm². Then, we multiply this by the height: 400 cm² * 125.66 cm = 50264 cm³. Finally, we multiply by π (approximately 3.14159): 50264 cm³ * 3.14159 ≈ 157913.6 cm³. Therefore, the volume of the cylinder is approximately 157913.6 cubic centimeters.
To better understand this volume, it's helpful to convert it to a more familiar unit, such as liters. Since 1 liter is equal to 1000 cubic centimeters, we can divide the volume in cubic centimeters by 1000 to get the volume in liters: 157913.6 cm³ / 1000 ≈ 157.9 liters. This means that our cylinder can hold approximately 157.9 liters of liquid. The volume calculation is a crucial step in understanding the cylinder's capacity and its potential applications. Whether it's a cylindrical tank storing water, a cylindrical container holding chemicals, or a cylindrical component in a machine, the volume is a key parameter that dictates its functionality. In our specific case, the volume of approximately 157.9 liters gives us a tangible sense of the cylinder's size and its potential use in various contexts. By calculating the volume, we have completed a comprehensive analysis of the cylinder's geometric properties, building upon our previous calculations of the radius and diameter. This holistic understanding of the cylinder's dimensions and capacity is essential for anyone working with cylindrical shapes in practical applications.
Conclusion
In summary, we have thoroughly explored the geometry of a cylinder under the specific condition that its height is equal to the circumference of its base, given a height of 125.66 cm. We began by establishing the fundamental principles of cylinder geometry, defining key parameters such as radius, diameter, and height, and their interrelationships. We then tackled the core problem of calculating the radius, employing the equation h = 2πr and arriving at an approximate radius of 20 cm. Building upon this result, we easily determined the diameter to be 40 cm, using the formula d = 2r. Finally, we calculated the volume of the cylinder using the formula V = πr²h, obtaining a volume of approximately 157913.6 cubic centimeters, or 157.9 liters.
This exercise has demonstrated the power of mathematical principles and formulas in dissecting and understanding geometric shapes. By systematically applying these tools, we were able to determine the key properties of the cylinder, even under a specific and somewhat unusual condition. The results we obtained – the radius, diameter, and volume – provide a comprehensive understanding of the cylinder's size, proportions, and capacity. Moreover, this exploration has highlighted the practical relevance of geometry in various fields, from engineering and architecture to everyday applications. The ability to calculate the dimensions and volume of cylindrical shapes is essential for anyone working with three-dimensional objects, whether it's designing a cylindrical tank, manufacturing a cylindrical component, or simply understanding the capacity of a cylindrical container. Therefore, the knowledge and skills gained in this article are valuable assets for anyone seeking to deepen their understanding of geometry and its real-world applications.