Cylinder, cylinder area

Cylinder (derived from the Greek language, from the words "roller", "roller") is a geometric body that is bounded from the outside by a surface called a cylindrical surface 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 motion of a straight line in space. These motions are such that the selected point of this straight line commutes along a curve of flat type. Such a straight line is called a generator, and a curved line is called a directrix.

The cylinder consists of a pair of bases and a lateral cylindrical surface. Cylinders are of several types:

1. A circular, straight cylinder. With such a cylinder the bases and the guide are perpendicular to the generatrix of the line, and there is an axis of symmetry.

2. Inclined cylinder. His angle between the forming line and the base is not straight.

3. The cylinder is of a different shape. Hyperbolic, elliptic, 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 lateral surface.

The formula by which the total area of the cylinder for a circular, straight cylinder is calculated:

Sp = 2n Rh + 2n R2 = 2n R (h + R).

The surface of the lateral surface is looked for a bit more complicated than the entire cylinder area, it is calculated by multiplying the length of the generatrix of the line by the perimeter of the section formed by the plane that is perpendicular to the generatrix of the line.

This cylinder surface area for a circular, straight cylinder is recognized by the sweep of this object.

A sweep is a rectangle that has a height h and a length P that is equal to the perimeter of the base.

It follows that the lateral area of the cylinder is equal to the sweep area and can be calculated from this formula:

Sb = Ph.

If we 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 must be equal to the product of the length of its generatrix and the perimeter of the section, which is perpendicular to the given line generator.

Unfortunately, there is no simple formula for expressing the area of the lateral surface of an inclined cylinder through its height and the parameters of its base.

To calculate the cross-sectional area of a cylinder, it is necessary to know several facts. If the section crosses 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 side of the axial section of the figure, which is perpendicular to the bases, is equal to the height, and the other to 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 to another, perpendicular to the first, or to 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 bottom of the cylinder, draw a radius and set aside the distance on which the section is located. And from this point it is necessary to draw perpendiculars to the radius from the intersection with the circle. The 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. Pythagoras' theorem sounds like this: "The sum of the squares of the 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 located as the area of the circle.

The area of the circle is:

S okr. = 2n R2.

To find the radius of the circle R, its length C must be divided by 2n:

R = C \ 2n, where n is the number pi, the mathematical constant calculated to work with the circle data and is equal to 3.14.