A cylinder is one of the spatial figures, which is the object of studying stereometry. An important characteristic of any figure is its surface area. In this article, we will consider what a cylinder scan is, and also show how to calculate its area.
Cylinder as a geometry figure
Suppose we have some smooth curve. It can be a circle, an ellipse, a parabola and so on. We take a segment of arbitrary length that does not lie in the plane of the curve, and use it to describe the surface, following the direction of the curve and translating the segment parallel to itself. The resulting surface is called a cylindrical or simply a cylinder. The marked curve is called the directrix (guide), and the segment is called the generalrix (generatrix).
If we add two more plane identical figures to the cylindrical surface, bounding this surface from the ends, then the resulting body is also called a cylinder. It consists of two equal bases and a cylindrical surface.
For a visual representation of the described figure, the following figure. It depicts an elliptical cylinder having half axes a and b and height h (the distance between the bases).
Round straight cylinder
Let the directrix be a circle of some radius. Then the figure formed by the generator will be called a round cylinder. Nevertheless, the generatrix relative to the plane bounded by the directrix can be directed arbitrarily. Therefore, in the general case, they speak of an inclined round cylinder. If the generatrix is perpendicular to the planes of the base, then the cylinder is called straight. It is shown in the figure below.
Here AB = DC is the radius of the cylinder, AD = BC = h is the height of the figure. For a straight cylinder, the height is always equal to the length of the generatrix. The segment AD indicates the axis of the cylinder - a straight line connecting the two central points of the bases. From the axis of the cylinder, all points of the lateral surface lie at the same distance equal to the radius of the figure.
Further in the article we will consider the development of the cylinder surface, round and straight.
Sweep round straight cylinder
When talking about the development of the cylinder, they mean the total surface area represented by flat figures. A round straight cylinder is formed by two circles having the same radii, and one side surface that connects these circles to each other.
The illustration shows a paper cylinder. How is the development of a cylinder sweep? Take imaginary scissors and cut one base along the circumference. Then we will do the same with the second base. We cut the side surface along the generatrix, that is, parallel to the axis of the figure, and expand it. Thus, we got a scan, which is shown in the figure below.
Obviously, it consists of two identical circles. As for the cylindrical surface, in a cut form it is represented by a rectangle. The scan is convenient to use when determining the surface area of the figure in question.
Cylinder sweep area
Above, we showed how you can get a scan of a figure. To calculate its area, it is necessary to add the area for all its parts, that is, for two circular bases and one rectangle.
We denote the radius of the base by the letter r, and the height of the figure by the letter h. The area of one base is equal to the area of the circle, that is:
S o = pi * r 2
Here pi is the Pi number, approximately equal to 3.14.
To calculate the area of a rectangle representing the lateral surface of a figure in expanded form, you need to know its two sides. One of them is equal to the height h. The second, as you might guess, corresponds to the length of the directrix, that is, the circumference. Denote it by l. Then we can write the following equalities:
l = 2 * pi * r;
S b = l * h = 2 * pi * r * h
Here S b is the area of the rectangle equal to the area of the cylindrical surface.
Given that the figure has two bases, add up the calculated values, we obtain the total area of the cylinder sweep:
S = 2 * S o + S b = 2 * pi * r 2 + 2 * pi * r * h = 2 * pi * r * (r + h)
The area S of the figure is uniquely determined through its radius and height.
We show how to use this equality to solve the geometric problem.
The task of finding the radius of the figure
It is known that the total surface of a round straight cylinder is 60 cm 2 . What is the radius of the base of the cylinder, if its height is 7 cm, and the figure has only one base.
The cylinder described in the problem statement is a barrel without a lid; therefore, its surface area is formed not by two, but by one base.
If we take into account the named fact, then the formula for the area of the figure is written in the following form:
S = pi * r 2 + 2 * pi * r * h
We substitute all the quantities known from the condition into this equality, we obtain:
60 = 3.14 * r 2 + 43.96 * r =>
3.14 * r 2 + 43.96 * r - 60 = 0
We got the classic full quadratic equation. Its solution will give the desired value of radius r. We solve through the discriminant:
D = 43.96 2 - 4 * 3.14 * (- 60) ≈ 2686.08;
r = (-43.96 ± √2686.08) / (2 * 3.14) = 1.25 cm
When solving the equation, the negative root was discarded, in view of its non-physical value.
Thus, the parameters of an open cylinder from the conditions of the problem are 7 cm in height and 2.5 cm in diameter.