A cylinder is one of the common forms of spatial bodies that we encounter daily. Indeed, a mug, tablet, chimney, pipe and other objects are cylindrical. In this article, we will consider the question of how to calculate the volume of a cylinder using various known parameters of this figure.
Defining a cylinder in geometry
Before proceeding to the answer to the question of how to calculate the volume of the cylinder, we will figure out which figure we are dealing with.
From a geometric point of view, the cylinder is formed by two one-dimensional elements. The first is the curve, which is the guideline. The second is a straight segment, which is called the generatrix. When the segment is not in the plane of the curve, if one of its ends is connected to the curve and moved parallel to itself along it, then we get a cylindrical surface.
Many spatial figures, including hyperbolic, parabolic and elliptical cylinders, are suitable for the given definition. Nevertheless, in this article we will consider only a round straight cylinder. It is called round because its bases are circles (the guide is a circle), and it is straight because the line segment is perpendicular to the bases. For clarity, the described cylinder is shown in the figure.
How to calculate cylinder volume through radius (diameter) and height?
The answer to this question is a standard formula that is valid for any cylinder and even a prism. We write it down:
V = S o * h
Since in this case the foundation is the right circle, you can specify this expression and rewrite it in the following form:
V = pi * r 2 * h
If the diameter is known, then you can find the volume of the cylinder using this expression:
V = pi / 4 * d 2 * h
Determination of cylinder volume through lateral surface area
Another way to calculate the volume of a cylinder is to use the area of ββits side surface. This surface is the set of points of all generators that connect the two bases of the figure. The side surface has a cylindrical shape. If you cut it along one of the generators and reveal it, you get a scan of the figure shown below.
It can be seen that in the expanded form, the side surface is an ordinary rectangle, the sides of which are equal to the height and circumference of the base. The latter fact allows us to write the formula for the area S b of this figure:
S b = 2 * pi * r * h
If the radius r of the figure is known, then its height will be equal to:
h = S b / (2 * pi * r)
Then, for volume V, the formula for the cylinder is written in the form:
V = r * S b / 2
If the area S b and the height h are known, then the radius of the figure will be equal to:
r = S b / (2 * pi * h)
Substituting it into the expression for volume, we arrive at the following formula:
V = S b 2 / (4 * pi * h)
You can see that both formulas using the lateral area S b correspond to the volume dimension (m 3 ).
It is important to understand that the volume of a round straight cylinder can be determined only if any two of its parameters are known.
The task of calculating the volume of the cylinder through the area of ββits full surface
Assume that the cylinder has a height of 21 cm and its sweep area is 335 cm 2 . It is necessary to determine the volume of the figure.
None of the above formulas can give us the desired answer. In this case, how to calculate the volume of the cylinder? As mentioned above, it is enough to know any two parameters of the figure to determine the value V. In this case, we first write the formula for the total area of ββthe cylinder:
S = S b + 2 * S o = 2 * pi * r * h + 2 * pi * r 2
We substitute the known data into this equality, we obtain:
r 2 + 21 * r - 53.34 = 0
After data substitution, we divided the left and right parts into 2 * pi and transferred all the members to one part of the equality.
Thus, we are faced with the task of solving the quadratic equation. We use the standard solution method through discriminant, we have:
discriminant D = 654.36;
r = 2.29 cm.
When solving the equation, we discarded the negative root.
Now, to determine the volume of the cylinder, you can use the formula with the parameters r and h. Substituting them into the indicated formula, we arrive at the answer to the problem: V = 345.8 cm 3 .