- 1 I. Formulas for calculating volumes of solids in space
- 2 II. Plane sections of solids
- 3 II. Enlargements and reductions
The course on the formulas of the volume of a solid and the study of sections of solids in space with situations of reduction or enlargement in third grade (3ème) is essential for the progression of the student.
The student will need to know the formulas by heart and know how to represent solids in space with the cavalier perspective or know how to draw his pattern. The student must also develop skills in the volume of solids by performing conversions of magnitudes or by applying the formulas for the volume of a cube, a right block, a cone or a pyramid.
We will finish this lesson on volumes and sections of solids with concrete examples from everyday life in the third grade.
I. Formulas for calculating volumes of solids in space
II. Plane sections of solids
1. section of a right block by a plane
The section of a right block by a plane (P) parallel to a face is an identical rectangle
to this face:
2. section of a cylinder of revolution by a plane
The section of a cylinder of revolution of radius R by a plane (P) parallel to the bases is a circle of radius R :
The section of a cylinder of revolution by a plane (P) parallel to the axis is a rectangle:
3.section of a pyramid by a plane
The section of a pyramid by a plane (P) parallel to the base is a polygon having the same shape as the base
4. section of a cone of revolution by a plane
The section of a cone of revolution by a plane (P) parallel to the base is a circle whose
the center belongs to the height of the cone.
II. Enlargements and reductions
When two figures have the same shape, we can calculate the following coefficient:
- if k >1, it is said to be an enlargement;
- if k <1, we say that it is a reduction.
In an enlargement or reduction of ratio k :
- the lengths are multiplied by k ;
- the areas are multiplied by ;
- volumes are multiplied by .
Consider a pyramid of volume undergoing an enlargement of ratio k=4
then the volume V’ after enlargement of this pyramid will be:.
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