Control on the sections of solids and volumes in third grade.

The sections of volumes of space are very important in geometry. So, with a math test on volumes and volume sections of space in third grade (3ème), you can make more progress.

Exercise 1:

We consider a cone of vertex S and base the disk of center O.

It is cut by a plane P parallel to the base, it is assumed that (OA) and (O’A’) are parallel.

The radii are given ;OA= 12cm and OA’=4,8 cm.

The distance between points O and O’ is 9 cm.

Cone of revolution

a) Calculate the measure of the angle \widehat{ASO} to the nearest degree.

b) Show that \frac{SO'}{SO}=\frac{2}{5}.

c) Show that SO =15 cm

d) Calculate the exact value of the volume of the cone of vertex S and base the disk of center O (as a function of \pi).

e) Calculate the volume of the cone with vertex S and base the disk with center O’.

Deduce the volume of the truncated cone to the nearest cm^3.

Exercise 2: Volume sections

We consider two vases (see figures below): one made of a regular pyramid and the other of a cylinder of revolution and both mounted on supports (grayed on the figure).


1. What is the nature of the quadrilateral ABCD ?

Calculate the volume V1 of vessel 1 in cm^3 and then in L.

2. The vase 1 is filled up to the middle (I is the middle of [SJ]), thus obtaining a “water pyramid”, reduction of the pyramid constituted by the vase.

a. What is the scale of this reduction?

b. By how much must the volume of the vase be multiplied to obtain the volume of the water?

c. Deduce that the remaining unoccupied volume represents the \frac{7}{8} of the initial volume V1.

3. We pour 512 cm^3 of water in the vase 2.

a. Noting V2 the volume of vessel 2, justify the fact that it does not overflow.

b. Calculate, to the nearest tenth, the height of water x in centimeters obtained in vase 2.

Exercise #3:

A cardboard box has the shape shown below:

  • ABCDA’B’C’D’ is a cube of 6 cm edge ;
  • SABCD is a regular pyramid whose four faces are equilateral triangles.

Cardboard box

1. Calculate the length AC.

2. Prove that the triangle SAC is right-angled.

3. Calculate the height SH of the pyramid SABCD.

Show that SH can be written 3\sqrt{2}.

4. Calculate the volume of the box rounded to the nearest cm^3.

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