A cylinder (comes from the Greek language, from the words "skating rink", "roller") is a geometric body that is bounded on the outside by a surface called a cylindrical and two planes. These planes intersect the surface of the figure and are parallel to each other.
A cylindrical surface is a surface that is obtained by translational movements of a straight line in space. These movements are such that the selected point of this straight line makes a movement along a curve of a flat type. Such a straight line is called a generatrix, and a curve line is called a guide.
The cylinder consists of a pair of bases and a lateral cylindrical surface. Cylinders come in several forms:
1. A circular, straight cylinder. In such a cylinder, the base and the guide are perpendicular to the generatrix line, and there is an axis of symmetry.
2. Inclined cylinder. He angle between the generatrix line and the base is not straight.
3. A cylinder of a different shape. Hyperbolic, elliptical, parabolic and others.
The area of the cylinder, as well as the total surface area of any cylinder, is found by adding the base areas of this figure and the area of the side surface.
The formula by which the total cylinder area is calculated for a circular, straight cylinder:
Sp = 2n Rh + 2n R2 = 2n R (h + R).
The lateral surface area is found a little more complicated than the entire cylinder area, it is calculated by multiplying the length of the generatrix line by the perimeter of the section formed by a plane that is perpendicular to the generatrix line.
This cylinder surface area for a circular, straight cylinder is recognized by the scan of this object.
A reamer is a rectangle that has a height h and a length P, which is equal to the perimeter of the base.
It follows that the lateral area of the cylinder is equal to the area of the sweep and can be calculated by this formula:
Sb = Ph.
If you take a circular, straight cylinder, then for him:
P = 2n R, and Sb = 2n Rh.
If the cylinder is inclined, then the area of the lateral surface should be equal to the product of the length of its generatrix line and the perimeter of the section, which is perpendicular to this generatrix line.
Unfortunately, there is no simple formula for expressing the lateral surface area of an inclined cylinder through its height and its base parameters.
To calculate the cross - sectional area of a cylinder, you need to know a few facts. If a section intersects the bases with its plane, then such a section is always a rectangle. But these rectangles will be different, depending on the position of the section. One of the sides of the axial section of the figure, which is perpendicular to the bases, is equal to the height, and the other is the diameter of the base of the cylinder. And the area of such a section, respectively, is equal to the product of one side of the rectangle on the other, perpendicular to the first, or the product of the height of this figure by the diameter of its base.
If the section is perpendicular to the base of the figure, but does not pass through the axis of rotation, then the area of this section will be equal to the product of the height of this cylinder and a certain chord. To get a chord, you need to build a circle at the base of the cylinder, draw a radius and set aside the distance at which the section is located. And from this point you need to draw perpendiculars to the radius from the intersection with the circle. Intersection points connect to the center. And the base of the triangle is the desired chord, the length of which is sought by the Pythagorean theorem. The Pythagorean theorem reads as follows: “The sum of the squares of two legs is equal to the hypotenuse squared”:
C2 = A2 + B2.
If the section does not affect the base of the cylinder, and the cylinder itself is circular and straight, then the area of this section is found as the area of a circle.
The circumference is:
S okr. = 2n R2.
To find the radius of a circle R, you need to divide its length C by 2p:
R = C \ 2n, where n is the number pi, the mathematical constant calculated for working with circle data and equal to 3.14.