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I. Formulas for areas of figures and volumes of solids :
1.area formulas of figures :
2.Form of the volumes of solids :
II. Plane sections of surfaces :
1. Section of a ball by a plane :
When the plane passes through the center of the ball, the section is a disk of the same center and radius.
2. section of a right block by a plane
3.section of a cylinder of revolution by a plane :
4.section of a pyramid by a plane :
same shape as the base.
5.Section of a cone of revolution by a plane :
III. Solid expansions and reductions:
When two figures have the same shape, we can calculate the following coefficient:
The reduction coefficient, noted k, is given by the formula:
Consider an enlargement (or reduction) of ratio k.
- If
then it is an enlargement;
- If
then it is a reduction.
When enlarging (or reducing) the ratio k :
- the lengths are multiplied by k ;
- the areas are multiplied by
;
- volumes are multiplied by
.
Example:
Consider the pyramid with base ABCD and the section IJKL made parallel to its base.
We know that SJ= 6 cm; SB = 10 cm; .
Calculate the area of the section IJKL.
The reduction coefficient is .
We have:
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