Volume – one of characteristics a body which is in space. For each type of spatial geometrical figures it is on the formula which is removed at summation of volumes of elementary figures.
It is required to you
- - concept about convex polyhedrons and bodies of rotation;
- - ability to calculate the area of polygons;
- - calculator.
Instruction
1. Find parallelepiped volume, using the fact that the relation of volumes of two parallelepipeds is equal to the relation of their heights. Consider three such figures which parties are equal to a, b, c; a, b, 1; a, 1.1. Where number 1 is the party of a single cube which is a standard of measurement of volume. Designate their volumes by V, V1 and V2. The parties which are on the third place, respectively will be heights. Take such ratios of volumes of parallelepipeds and a cube of V/V1=c/1; V1/V2=b/1; V2/1=a/1. Then term by term multiply the left and right parts. Receive V/V1•V1/V2 • V2/1=a • b•c. Make reduction and receive V=a • b•c. The volume of a parallelepiped is equal to the work of its linear sizes. In this way it is possible to remove formulas for calculation of volumes and for other solids.
2. To determine the volume of any prism, find the area of its basis of Sosn, and increase h by its height (V=Sosn·h). Take the piece which is carried out from one of tops perpendicular to the plane of other basis for height of a prism.
3. Example. Determine the volume of a prism in which basis the square with the party of 5 cm lies, and height is 10 cm. Find the area of the basis. As it is a square, Sosn =5? =25 cm?. Find the volume of a prism of V=25•10=250 cm?.
4. For scoping of a pyramid find its area of its basis and height. Then 1/3 increase by this Sosn square and at h height (V=1/3·Sosn·h). Height represents the piece lowered from top is perpendicular to the basis plane.
5. Example. The equilateral triangle with the party of 8 cm is the cornerstone of a pyramid. Its height is equal to 6 cm. Determine its volume. As in the basis the equilateral triangle lies, define its square as the work of a square of the party at a root from 3 divided into 4. Sosn =v3•8?/4=16v3 cm?. Determine volume by formula V=1/3 • 16v3 • 6=32v3? 55.4 cm?.
6. For a cylinder use the same formula, as for V=Sosn prism • h, and for a cone – for a pyramid V=1/3·Sosn·h. To find sphere volume, learn its radius of R, and use formula V=4/3 •? • R?. When calculating consider that?? 3.14.