When studying stereometry, one of the main topics becomes the “Cylinder”. The lateral surface area is considered, if not the main, then an important formula in solving geometric problems. However, it is important to remember the definitions that will help to orient in the examples and in the proof of various theorems.
Cylinder concept
First, you need to consider a few definitions. Only after studying them can we begin to consider the question of the formula for the area of the lateral surface of the cylinder. Based on this record, other expressions can also be calculated.
- A cylindrical surface is understood to mean a plane described by a generator moving and remaining parallel to a given direction, sliding along an existing curve.
- There is a second definition: a cylindrical surface is formed by many parallel lines intersecting a given curve.
- Generator is called conditionally the height of the cylinder. When it moves around an axis passing through the center of the base, a marked geometric body is obtained.
- An axis means a straight line passing through both bases of a figure.
- A cylinder is a stereometric body bounded by intersecting lateral surfaces and 2 parallel planes.
There are varieties of this three-dimensional figure:
- By circular is meant a cylinder whose guide is a circle. Its main components are the radius of the base and the generatrix. The latter is equal to the height of the figure.
- There is a straight cylinder. It got its name due to the perpendicularity of the figure forming to the bases.
- The third type is a beveled cylinder. In the textbooks you can also find its other name "circular cylinder with a beveled base." This figure is determined by the radius of the base, the minimum and maximum heights.
- An equilateral cylinder is understood to mean a body having equal height and diameter of a circular plane.
Legend
Traditionally, the main "components" of the cylinder are usually called as follows:
- The radius of the base is R (it also replaces the same magnitude of the stereometric figure).
- Generating - L.
- Height - H.
- The base area is S main (in other words, it is necessary to find the indicated circle parameter).
- The height of the beveled cylinder is h 1 , h 2 (minimum and maximum).
- The area of the side surface is S side (if you expand it, you get a kind of rectangle).
- The volume of the stereometric figure is V.
- Full surface area - S.
"Components" of a stereometric figure
When a cylinder is studied, the lateral surface area plays an important role. This is due to the fact that this formula is included in several other, more complex ones. Therefore, you must be well-versed in theory.
The main components of the figure are:
- Side surface. As you know, it is obtained due to the motion of the generator along a given curve.
- The full surface includes the existing base and side plane.
- The section of the cylinder, as a rule, is a rectangle parallel to the axis of the figure. Otherwise, it is called a plane. It turns out that the length and width in combination are the components of other figures. So, conditionally the lengths of the section are generators. Width - parallel chords of a stereometric figure.
- By axial section is meant the location of the plane through the center of the body.
- And finally, the final definition. A tangent is a plane passing through a cylinder generatrix and at right angles to the axial section. In this case, one condition must be fulfilled. The specified generatrix must be included in the plane of the axial section.
Basic formulas for working with a cylinder
In order to answer the question of how to find the surface area of the cylinder, it is necessary to study the basic "components" of the stereometric figure and the formulas for their location.
These formulas differ in that, first, expressions are given for a beveled cylinder, and then for a straight cylinder.
Case Studies
Task 1
It is necessary to find out the area of the lateral surface of the cylinder. Given the diagonal section AC = 8 cm (moreover, it is axial). In contact with the generatrix, < ACD = 30 °
Decision. Since the values of the diagonal and angle are known, then in this case:
A comment. The ACD triangle, in the specific example, is rectangular. This means that the quotient of the division of CD and AC = the cosine of the existing angle. The meaning of trigonometric functions can be found in a special table.
Similarly, you can find the value of AD:
Now it is necessary to calculate the desired result according to the following formulation: the area of the lateral surface of the cylinder is equal to twice the result of multiplying “pi”, the radius of the figure and its height. Another formula should be used: the area of the base of the cylinder. It is equal to the result of multiplying pi by the square of the radius. And finally, the last formula: total surface area. It is equal to the sum of the previous two areas.
Task 2
Cylinders are given. Their volume = 128 * p cm³. Which cylinder has the smallest full surface?
Decision. First you need to use the formulas to find the volume of the figure and its height.
Since the total surface area of the cylinder is known from theory, it is necessary to apply its formula.
If we consider the resulting formula as a function of the area of the cylinder, then the minimum "indicator" will be achieved at the point of extremum. To obtain the last value, it is necessary to use differentiation.
Formulas can be found in a special table for finding derivatives. In the future, the result found is equal to zero and the solution of the equation is found.
Answer: S min will be achieved at h = 1/32 cm, R = 64 cm.
Task 3.
A stereometric figure is given — a cylinder and a section. The latter is carried out in such a way that it is parallel to the axis of the stereometric body. The cylinder has the following parameters: VK = 17 cm, h = 15 cm, R = 5 cm. It is necessary to find the distance between the section and the axis.
Decision.
Since by the section of the cylinder we mean the SCMS, i.e., the rectangle, its side BM = h. It is necessary to consider the IUD. The triangle is rectangular. Based on this statement, we can deduce the correct assumption that MK = BC.
VK² = BM² + MK²
MK² = BK² - VM²
MK² = 17² - 15²
MK² = 64
MK = 8
From this we can conclude that MK = BC = 8 cm.
The next step is to draw the section through the base of the figure. It is necessary to consider the resulting plane.
AD is the diameter of the stereometric figure. It is parallel to the cross section mentioned in the condition of the problem.
BC is a straight line located on the plane of an existing rectangle.
ABCD is a trapezoid. In a particular case, it is considered to be isosceles, because a circle is described around it.
If you find the height of the resulting trapezoid, then you can get the answer set at the beginning of the task. Namely: finding the distance between the axis and the section taken.
To do this, find the values of AD and OS.
Answer: the cross section is 3 cm from the axis.
Tasks for fixing material
Example 1
The cylinder is given. The lateral surface area is used in a further solution. Other parameters are known. The base area is Q, the axial section area is M. It is necessary to find S. In other words, the total area of the cylinder.
Example 2
The cylinder is given. The lateral surface area must be found in one of the steps of solving the problem. It is known that height = 4 cm, radius = 2 cm. It is necessary to find the total area of the stereometric figure.