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A series of exercises on volumes and sections in 3rd grade on **geometry in space **allows you to discover a new method of calculation and to progress in math.

These exercises in the third grade involve the following concepts:

- the different volumes: ball, pyramid, cone of revolution, right prism;
- formula for calculating volumes;
- sections of volumes in
- reduction and enlargement.

Sections of solids in space are geometric figures obtained by cutting a solid with a plane. These figures can be polygons (such as a triangle or quadrilateral), circles or ellipses. Different sections can be obtained by choosing different cutting planes for the same solid.

These exercises are available for free download in PDF format.

## Exercise 1: ‘volumes

We realize the section of a pyramid SABCD with rectangular base by a parallel plane

at its base at 5 cm from the top.

We have AB=4.8 cm; BC = 4.2 cm and SH = 8 cm.

- Calculate the volume of the pyramid SABCD.
- The pyramid SA’B’C’D’ is a reduction of the pyramid SABCD.
- Give the ratio of this reduction.
- Deduce the volume of the pyramid SA’B’C’D’.

## Exercise 2:

A restaurant offers three scoops of ice cream for dessert

assumed perfectly spherical, diameter 4.2 cm.

The chocolate ice cream tub is a rectangular parallelepiped and is full,

as well as the cylindrical vanilla ice cream jar.

The restaurant owner wants to make cups with two chocolate balls and one vanilla ball.

- Show that the volume of a jar of chocolate ice cream is 3600 .

2. calculate the volume of a jar of vanilla ice cream to the nearest .

3. calculate the value of the volume of a ball contained in the cup, rounded to the nearest .

4) Knowing that the restaurant owner must make 100 cups of ice cream, how much must he buy?

of chocolate and vanilla pots?

## Exercise 3:

A regular pyramid with vertex S and height SO = 9 cm has

base a square ABCD of side 4,5 cm.

It is cut by a plane parallel to its base which intersects its height at O, such that SO’ = 6 cm.

a. What is the nature of the section A’B’C’D’?

b. Calculate the volume of the reduced pyramid SA’B’C’D’.

## Exercise 4: ‘volumes

A cone of height SO = 18 cm has for base a disk of radius 15 cm.

A is the point on the height [SO] such that SA=10 cm.

The plane passing through A and parallel to the base intersects the segment [SM] at N.

Calculate the volume, in , of the reduced cone of vertex S and base the disk of center A and radius AN.

Give an approximate value to the nearest unit.

## Exercise 5:

The blue rectangle is the section of the right prism ABCDEF by a plane parallel to the face BCFE and

passing through a point M of the edge [AB].

Give the nature and dimension(s) of this section:

a. when M is in A,

b. when M is in B,

c. when M is the middle of [AB].

## Exercise 6:

By cutting this rectangular parallelepiped by the plane passing through A and C and parallel to the edge [DH],

we obtain the AEGC section.

a. What is its nature?

b. Draw this section at full size.

c. Calculate the length AC, in cm.

## Exercise 7:

ABCDEFGH is a rectangular parallelepiped such that AB = 12 cm, AD=5 cm, AE=8 cm.

The point M of [AE] is such that AM =6 cm.

The section of this solid by a plane parallel to the face ABCD and passing through M is represented in blue.

a. Use the Pythagorean theorem to calculate AC.

b. Use Thales’ theorem to calculate MN.

## Exercise 8:

To create a decorative lamp, a machine slices a metal cylinder according to the

data indicated.

Calculate the exact value of MN.

## Exercise 9: ‘volumes

SABCD is a regular pyramid with square base of side 6 cm and height P

[SO] with SO = 7.5 cm.

A plane parallel to the base intersects [SO] at I so that SI=2.5 cm.

The section is the quadrilateral MNPQ.

a. Calculate the volume , in , of SABCD.

b. is the volume, in , of SMNPQ.

Express as a function of .

Give an approximate value of to the nearest hundredth.

## Exercise 10:

A cone of revolution with vertex S and base a disk with center O is cut by a plane

parallel to its base.

The section is the circle of center I which passes through B, point of intersection of the segment [SA] with the plane.

a. The cone of vertex S and whose base is the disk of center I passing through B is a reduction of the cone .

Write the reduction ratio in three different ways.

b. We give SO =10 cm, 0A=7.5 cm and SI=6 cm.

Draw the section at full size.

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