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’.
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?
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.
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].
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.
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.
To create a decorative lamp, a machine slices a metal cylinder according to the
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.
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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