sommaire
The volume of a solid is the measure of the space it occupies in three-dimensional space. It is usually expressed in units of cubic length, such as the cubic meter (
To develop skills in drawing a cylinder or a right block with a cavalier perspective and also to represent the pattern of different solids in space.
I. The right prism and volumes of solids
1. vocabulary
A right prism is a solid of space having its two bases which are superimposable polygons and its side faces are rectangles.
- The base of the first prism is a triangle
- It has five faces including three sides, 9 edges and six vertices.
- The base of the second prism is a pentagon.
- It has 7 faces including 5 sides, 15 edges and 10 vertices.
Remark:
- All side faces have a common dimension: the height of the prism.
- The number of side faces is equal to the number of sides of the base.
2. Pattern of a right prism
Example:
Here is the pattern of a right prism. Its base is a triangle whose sides are 5 cm, 4 cm and 3 cm long and whose height is 2 cm.
II. The cylinder of revolution
A cylinder of revolution is a solid with two bases that are superimposable disks and the lateral surface is a rectangle wrapped around the bases.
- The two bases are disks of the same radius.
- The line that joins the centers of the two bases is called the cylinder axis.
- The height of the cylinder is the length of the segment that joins the centers of the two base disks.
2. Pattern of a cylinder of revolution
Example:
Here is the pattern of a cylinder of revolution of height 3 cm having for base a disk of radius 1 cm.
The lateral surface of this cylinder is a rectangle:
- whose width is the height of the prism, i.e. 3 cm.
- whose length is the perimeter of the base disk, i.e.
.
III. Sections of solids and volumes
The section of a solid by a plane is the intersection between the solid and the plane.
The section of a prism by a plane parallel to a base is a polygon identical to the base.
Example:
A prism with a triangular base is cut by a plane parallel to its base.
The section is a triangle identical to the base triangle.
Remark:
Pavers are particular prisms, for which the section of a plane parallel to the base is a rectangle identical to this base.
Example:
We cut a cylinder of revolution of height 4 cm whose base radius is 1 cm, by a plane perpendicular to its axis.
The section is a disk of radius 1 cm.
The section of a cylinder of revolution by a plane which is perpendicular to its axis of rotation is a disk having the same radius as the base of this cylinder of revolution.
The section of a cylinder of revolution by a plane containing its axis of rotation is a rectangle.
Example:
A cylinder of revolution of height 5 cm, whose base radius is 2 cm, is cut by a plane containing its axis.
The section is a rectangle of length the height of the cylinder: 5 cm and width the diameter of the base: 4 cm.
IV: Calculation of solid volumes
To calculate the volume V of a right prism or a cylinder of revolution, we multiply the area of a base by its height h.
Example:
An attic has the shape of a right prism with a triangular base. We want to calculate its volume.
We calculate the area of a base which is a right triangle:
We multiply the area of a base by the height :
The volume of this attic is 30 m².
A can has the shape of a cylinder of revolution.
We want to calculate its capacity in centilitres.
We calculate the area of a base which is a disk of radius 3 cm.
We multiply the area of a base by its height which is 11 cm.
.
The volume of this can is approximately 311 or 311 mL or 31.1 cL.
Skills to be assimilated on volumes of solids:
- Know the definitions of the cube, right block, right prism and cylinder;
- Apply the formulas to calculate the volumes of the cube, the right parallelepiped, the right prism and the cylinder.
- Perform volume conversions;
- Know how to represent the pattern of a solid in space.
This course is in accordance with the officialnational education programs.
As a complement, you can consult the exercises on the volumes of solids in fifth grade.
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