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 () or the liter (). The volume of a solid can be calculated using specific formulas for different geometric shapes, such as the cube, cylinder, cone, etc. It can also be measured by filling an appropriately shaped container with units of volume such as marbles or cubes.
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
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.
- 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
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
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.
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.
Pavers are particular prisms, for which the section of a plane parallel to the base is a rectangle identical to this base.
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.
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.
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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