The pyramid and the cone are a very important chapter in eighth grade math. The student will need to know all the definitions as well as the different vocabulary regarding faces, vertex, base and height. Thus, it must also develop skills in calculations with the formulas of volume:

## I. The pyramid

1. vocabulary

Definitions:

It is a solid whose face in contact with the ground is called the base which is a polygon, the other faces are called the side faces and are triangles that have a common vertex, called the top of the pyramid.

Its height is the distance between its top and its base.

A side edge is a segment joining the top of the pyramid and one of the vertices of its base.

Example:
• The top of this pyramid is the S point.
• The base of this pyramid is the pentagon ABCDE.
• The side faces are the triangles SAB, SBC, SCD, SDE, SEA.
• The lateral edges are the segments [AS], [BS], [CS], [DS], [ES].
• The height of the pyramid is the segment [OS].
Remarks:
• A pyramid with a triangular base is called a tetrahedron.

• A regular pyramid has a base that is a regular polygon (for example an equilateral triangle or a square) and whose faces are isosceles triangles that can be superimposed. Its height passes through the center of the base which is the point of intersection of the diagonals.

2. pattern of a pyramid

Example:
Here is the pattern of a pyramid.
Its base is a rectangle, length 9 cm and width 6 cm, and each side edge measures 7 cm.

## II. The cone of revolution

1. vocabulary

Definition:

• A cone of revolution is a solid generated by a right-angled triangle rotating about an axis that is one of the sides adjacent to the right angle of the right-angled triangle.
• The base of a cone of revolution is a disk.
• The height of a cone of revolution is the distance between its top and its base.
• A generatrix of a cone of revolution is a segment that joins the vertex of the cone of revolution and a point of the circle of its base.
Example:
• The apex of the cone is the point S.
• The base of this cone is the disk of center O: it is represented in perspective by an oval (an ellipse) because it is not seen from the front.
• The height of the cone is the segment [OS].
• The triangle AOS, rectangular in O, generates the cone rotating around (OS).
• A generatrix of the cone is [SA].

2. pattern of a cone of revolution

Here is the pattern of a cone of revolution with base radius 3 cm and generatrix 5 cm.

The length of the sector of disk of radius 5 cm is equal to the perimeter of the base that is cm.

The angle of the disk sector is proportional to its length. It has the angle =36×6=216 °.

## IV. Volume calculations

The cone of revolution and the pyramid are “pointed” solids.

Ownership:

The volume is given by the following formula:

Remark:

When lengths are expressed in m, the area of the base is expressed in m², and the volume in .

Applications:

I. The pyramid of the Louvre is a regular pyramid with a square base of 35 m on each side, its height is 22 m.
1. Calculate the area of its base.
2. Calculate the exact value of the volume V of this pyramid.
Give the rounded value of V per cubic meter.
3. In an amusement park, a reduction of this pyramid is built; the side of the square base measures 7 m.
a. Calculate the scale of this reduction.
b. Calculate the height of the reduced pyramid.
c. By what number must the volume V of the Louvre pyramid be multiplied to obtain the volume V’ of the reduced pyramid?

II. We give: AB = 6 m, AE = 5 m, AD = 1.80 m, BC = 0.80 m .
On the diagram above, the dimensions are not respected.
1. Show that the volume of this pool is 39 .
2. At the end of the summer, Mr. OBAMA empties his pool with a
pump with a flow rate of 5 per hour.

Calculate the number of remaining in the pool after 5 hours.

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