The cylindrical geometrical form is used by production of automobile engines, other technical and household devices, and not only. To determine the area of a cylinder, it is necessary to find its full surface.
Instruction
1. According to Euclid's definition, cylinder it is formed in space as a result of rotation of a rectangle. Other mathematician, Cavalieri, gave to this figure more generalized definition in the form of rotation of the forming straight line. Rotation happens on some guide of the line which, in the simplest case, is a circle. However any closed figure can be the basis of a cylinder.
2. The bases are always parallel each other and equal. Moreover, any two cross sections and also the forming pieces have these properties. To determine the area of a cylinder, it is necessary to use a formula: S = Sb + 2•So where Sb is the area of a side surface, So is the area of the basis.
3. If to develop the simplest, circular cylinder on a rotation axis, then the rectangle with the parties equal to perimeter of the basis and height of a cylinder will turn out. According to a formula of the area of this two-dimensional figure, it is equal to the work of length of the basis on height. Therefore, the area of a side surface of a cylinder represents result of multiplication of perimeter of the basis by height: Sb = Ro·h.
4. The considered rectangle and two circles of the basis are called development of a cylinder. This term is applied during creation of technical drawings. The perimeter of a circle is equal to the double work of its radius on number π from where: Sb = 2 • π\• R•h.
5. It was necessary to find the areas of the bases of a cylinder. They are also connected with number π and depend on R radius: So = π\• R².
6. Substitute sizes in a basic formula: S = 2 · π\• R · h + 2 • π\• R² = 2 • π\• R • (h + R).
7. At the generalized cylinder the directing line is a broken line, and the corresponding cylindrical surface can be presented in the form of a number of the rectangles formed by couples of parallel forming straight lines. Sections in this case are polygons, and the area of such cylinder is defined similar to the area of a full surface of a prism.