The cylinder is one of the simple three-dimensional figures, which is studied in the school course of geometry (section stereometry). In this case, often problems arise in calculating the volume and mass of the cylinder, as well as in determining the area of its surface. Answers to these questions are given in this article.
What is a cylinder?
Before proceeding to answer the question of what is the mass of the cylinder and its volume, it is worth considering what this spatial figure is. It should be noted right away that a cylinder is a three-dimensional object. That is, in space, you can measure its three parameters on each axis in a Cartesian rectangular coordinate system. In fact, to uniquely determine the dimensions of the cylinder, it is enough to know only two of its parameters.
A cylinder is a three-dimensional figure formed by two circles and a cylindrical surface. To make this object clearer, it’s enough to take a rectangle and begin to rotate it around any of its sides, which will be the axis of rotation. In this case, the rotating rectangle will describe the rotation figure - the cylinder.
Two round surfaces are called the bases of the cylinder, they are characterized by a certain radius. The distance between the bases is called height. Two bases are interconnected by a cylindrical surface. A line passing through the centers of both circles is called the axis of the cylinder.
Volume and surface area
As can be seen from the above, a cylinder is determined by two parameters: height h and radius of its base r. Knowing these parameters, one can calculate all other characteristics of the body under consideration. The following are the main ones:
- The area of the grounds. This value is calculated by the formula: S 1 = 2 * pi * r 2 , where pi is the number pi equal to 3.14. The number 2 in the formula appears because the cylinder has two identical bases.
- The area of the cylindrical surface. It can be calculated as follows: S 2 = 2 * pi * r * h. To understand this formula is simple: if you cut a cylindrical surface vertically from one base to another and turn it around, you will get a rectangle whose height will be equal to the height of the cylinder, and the width will correspond to the circumference of the base of the volumetric figure. Since the area of the resulting rectangle is the product of its sides, which are equal to h and 2 * pi * r, we get the above formula.
- The surface area of the cylinder. It is equal to the sum of the areas S 1 and S 2 , we obtain: S 3 = S 1 + S 2 = 2 * pi * r 2 + 2 * pi * r * h = 2 * pi * r * (r + h).
- Volume. This value is simple, it is only necessary to multiply the area of one base by the height of the figure: V = (S 1/2 ) * h = pi * r 2 * h.
Determination of cylinder mass
Finally, it’s worth going directly to the topic of the article. How to determine the mass of the cylinder? To do this, you need to know its volume, the formula for calculating which was presented above. And the density of the substance of which it consists. Mass is determined by a simple formula: m = ρ * V, where ρ is the density of the material forming the object in question.
The concept of density characterizes the mass of a substance that is in a unit volume of space. For instance. Iron is known to have a higher density than wood. This means that in the case of equal volumes of iron and wood, the first will have a much larger mass than the second (approximately 16 times).
Calculation of the mass of the copper cylinder
Consider a simple task. It is necessary to find the mass of the cylinder made of copper. For definiteness, let the cylinder have a diameter of 20 cm and a height of 10 cm.
Before you begin to solve the problem, you should understand the source data. The radius of the cylinder is equal to half its diameter, which means r = 20/2 = 10 cm, the height is h = 10 cm. Since the cylinder considered in the problem is made of copper, then referring to the reference data, we write out the density value of this material: ρ = 8 , 96 g / cm 3 (for a temperature of 20 ° C).
Now you can begin to solve the problem. To begin with, we calculate the volume: V = pi * r 2 * h = 3.14 * (10) 2 * 10 = 3140 cm 3 . Then the mass of the cylinder will be equal to: m = ρ * V = 8.96 * 3140 = 28134 grams or about 28 kilograms.
You should pay attention to the dimensionality of units during their use in the corresponding formulas. So, in the task, all parameters were presented in centimeters and grams.
Homogeneous and hollow cylinders
From the result obtained above, it can be seen that a copper cylinder with relatively small dimensions (10 cm) has a large mass (28 kg). This is due not only to the fact that it is made of heavy material, but also to the fact that it is homogeneous. This fact is important to understand, since the above formula for calculating the mass can be used only if the cylinder completely (outside and inside) consists of the same material, that is, it is homogeneous.
In practice, hollow cylinders (for example, cylindrical water barrels) are often used. That is, they are made of thin sheets of some material, but inside are empty. For a hollow cylinder, the specified mass calculation formula cannot be used.
Calculation of the mass of the hollow cylinder
It is interesting to calculate what mass a copper cylinder will have if it is empty inside. For example, let it be made of a thin copper sheet with a thickness of only d = 2 mm.
To solve this problem, you need to find the volume of the copper itself from which the object is made. And not the volume of the cylinder. Since the thickness of the sheet is small compared to the dimensions of the cylinder (d = 2 mm and r = 10 cm), then the volume of copper from which the item is made can be found by multiplying the entire surface area of the cylinder by the thickness of the copper sheet, we obtain: V = d * S 3 = d * 2 * pi * r * (r + h). Substituting the data from the previous problem, we obtain: V = 0.2 * 2 * 3.14 * 10 * (10 + 10) = 251.2 cm 3 . The mass of the hollow cylinder can be obtained by multiplying the obtained volume of copper, which was required for its manufacture, by the density of copper: m = 251.2 * 8.96 = 2251 g or 2.3 kg. That is, the considered hollow cylinder weighs 12 (28.1 / 2.3) times less than a homogeneous one.