Planar sections of solids: 3rd grade math class.

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1 I. Formulas for areas of figures and volumes of solids :
- 1.1 1.area formulas of figures :
- 1.2 2.Form of the volumes of solids :
2 II. Plane sections of surfaces :
3 III. Solid expansions and reductions:

A 9th grade math course on volumes of solids and sections of solids in space. In this lesson, we will look at the area of figures (rectangle, parallelogram, trapezium) and the formulas for calculating the volume of a pyramid, a cylinder of revolution or a ball. Then, in a second step, we will make sections of solids by a plane and we will make calculations with the concepts of reduction and enlargement.

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 :

Definition:

In geometry, the intersection between a solid and a plane is called a plane section.

1. Section of a ball by a plane :

Ownership:

The section of a ball through a plane is a disk.
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

Ownership:

The section of a right block by a plane parallel to a face is a rectangle.

Ownership:

The section of a right block by a plane parallel to an edge is a rectangle.

3.section of a cylinder of revolution by a plane :

Ownership:

The section of a cylinder of revolution of radius R by a plane parallel to the bases is a disk of radius R.

4.section of a pyramid by a plane :

Ownership:

The section of a pyramid by a plane parallel to the base is a polygon having the
same shape as the base.

5.Section of a cone of revolution by a plane :

Ownership:

The section of a cone of revolution by a plane parallel to the base is a disk whose center belongs to the height of this cone.

III. Solid expansions and reductions:

Definition:

Consider a plane section parallel to a base.

We obtain a reduction (or an enlargement) of the solid.
When two figures have the same shape, we can calculate the following coefficient:
The reduction coefficient, noted k, is given by the formula: $k=\frac{longueur\,finale}{longueur\,\,initiale}$

>0.

Ownership:

Consider an enlargement (or reduction) of ratio k.

If then it is an enlargement;
If then it is a reduction.

Ownership:

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; $A_{ABCD}=24_,cm^2$ .

Calculate the area of the section IJKL.

The reduction coefficient is $k=\frac{SJ}{SB}=\frac{6}{10}=0,6<1$ .

We have:

$A_{IJKL}=k^2\times \,A_{ABCD}\\A_{IJKL}=0,6^2\times \,24\,\\A_{IJKL}=8,64\,cm^2$

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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 :

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 :

5.Section of a cone of revolution by a plane :

III. Solid expansions and reductions:

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