A cylinder is a symmetrical spatial figure whose properties are considered in high school with a course on stereometry. For its description, linear characteristics such as the height and radius of the base are used. In this article, we will consider questions regarding what an axial section of a cylinder is and how to calculate its parameters through the basic linear characteristics of a figure.
Geometric figure
First, we give a definition to the figure, which will be discussed in the article. A cylinder is a surface formed by parallel movement of a segment of a fixed length along a certain curve. The main condition for this movement is that the segment of the plane of the curve should not belong.
The figure below shows a cylinder whose curve (guide) is an ellipse.
Here, a segment of length h is its generatrix and height.
It can be seen that the cylinder consists of two identical bases (ellipses in this case), which lie in parallel planes, and the side surface. The last belongs to all points of the generating lines.
Straight and tilted cylinders
Before proceeding to the consideration of the axial section of the cylinders, we will describe what types of these figures are.
If the generatrix is perpendicular to the base of the figure, then we speak of a straight cylinder. Otherwise, the cylinder will be inclined. If you connect the center points of two bases, then the resulting line is called the axis of the figure. The figure below shows the difference between straight and tilted cylinders.
It can be seen that for a straight figure, the length of the generating segment coincides with the value of height h. For an inclined cylinder, the height, i.e. the distance between the bases, is always less than the length of the generatrix line.
Next, we describe the axial sections of both types of cylinders. In this case, we will consider the figures, the basis of which is a circle.
Axial section of a straight cylinder
Axial is any section of a cylinder that contains its axis. This definition means that the axial section will always be parallel to the generatrix.
In the cylinder, the straight axis passes through the center of the circle and is perpendicular to its plane. This means that the circle under consideration will intersect in its diameter. The figure shows the half of the cylinder, which was obtained as a result of the intersection of the figure with a plane passing through the axis.
It is not difficult to understand that the axial section of a straight circular cylinder is a rectangle. Its sides are the diameter d of the base and the height h of the figure.
We write the formulas for the axial sectional area of the cylinder and the length h d of its diagonal:
S = h * d;
h d = √ (h 2 + d 2 )
A rectangle has two diagonals, but both are equal to each other. If the radius of the base is known, then it is not difficult to rewrite these formulas through it, given that it is half the diameter.
Axial section of an inclined cylinder
The figure above shows an inclined cylinder made of paper. If you perform its axial section, then you get no longer a rectangle, but a parallelogram. Its sides are known quantities. One of them, as in the case of a straight cylinder section, is equal to the diameter d of the base, while the other is the length of the generatrix segment. Denote it by b.
To unambiguously determine the parameters of a parallelogram, it is not enough to know its side lengths. An angle between them is also needed. Assume that the acute angle between the guide and the base is α. It will be the angle between the sides of the parallelogram. Then the formula for the axial sectional area of the inclined cylinder can be written as follows:
S = d * b * sin (α)
The diagonals of the axial section of an inclined cylinder are somewhat more difficult to calculate. A parallelogram has two diagonals of different lengths. We present without derivation the expressions that allow us to calculate the diagonals of a parallelogram along the known sides and the acute angle between them:
l 1 = √ (d 2 + b 2 - 2 * b * d * cos (α));
l 2 = √ (d 2 + b 2 + 2 * b * d * cos (α))
Here l 1 and l 2 are the lengths of the small and large diagonals, respectively. These formulas can be obtained independently if we consider each diagonal as a vector by introducing a rectangular coordinate system on the plane.
Direct cylinder problem
We show how to use the acquired knowledge to solve the following problem. Let a round straight cylinder be given. It is known that the axial section of a cylinder is a square. What is the area of this section equal if the surface area of the whole figure is 100 cm 2 ?
To calculate the required area, it is necessary to find either the radius or the diameter of the base of the cylinder. To do this, we use the formula for the total area S f of the figure:
S f = 2 * pi * r * (r + h)
Since the axial cross section is a square, this means that the radius r of the base is half the height h. Given this, we can rewrite the equality above in the form:
S f = 2 * pi * r * (r + 2 * r) = 6 * pi * r 2
Now we can express the radius r, we have:
r = √ (S f / (6 * pi))
Since the side of the square section is equal to the diameter of the base of the figure, the following formula will be valid for calculating its area S:
S = (2 * r) 2 = 4 * r 2 = 2 * S f / (3 * pi)
We see that the desired area is uniquely determined by the surface area of the cylinder. Substituting the data into equality, we come to the answer: S = 21.23 cm 2 .