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**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 :

**solid**and a plane is called a plane

**section**.

### 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:

**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: >0.

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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