Proper 6 coal pyramid. The volume of the hexagonal pyramid. Protection of personal information
Hexagonal pyramid A polyhedron is called, at the base of which the correct hexagon is lying, and the side faces are formed by the same inaccessible triangles.
Such pyramids have many unique properties:
- All sides of the base of the same length;
- All side ribs are equal to each other;
- All the angles at the base are equal, as well as dwarfrani corners formed by the ribs equal;
- Each side face of the same area.
It is calculated from the area of \u200b\u200bits base and side sweep. To calculate the volume it is enough to know the pyramid height and its base area. To begin with, we'll figure it out with the reference formula of the right hexagon.
One of the most significant differences between the correct hexagon from the remaining figures is the equality of its side by the radius of the circumference of the circumference. Due to this property, the base area of \u200b\u200bthe correct hexagonal pyramid is calculated by the formula: 
To calculate, you can use both the radius of the circle described and the length of the side of the right hexagon.
Now back to the formula of the hexagonal pyramid. It represents one third of the production of the ground area to the height of the pyramid, lowered to this basis:
Now consider an example of calculating the volume of the hexagonal pyramid.
Let the correct hexagonal pyramid be given, the height of which is H \u003d 8 cm. Around the base is described a circle with a radius R \u003d 6 cm. Find the volume.
In the calculation of the required parameter there will be nothing complicated - after all, all the necessary values \u200b\u200bare specified by the conditions. Therefore, we find the area of \u200b\u200bthe foundation of our polyhedron. Remember that the radius of the circumference described around the correct hexagon is equal to its parties. Substitute data in the formula:
Now we can use the found area for calculating the volume of our hexagonal pyramid: 
Thus, knowing the properties of the right hexagon and the volume formula for the hexagonal pyramid, we found all the necessary parameters.
The pyramid, at the base of which the right hexagon lies, and the side parties are formed by the right triangles, called hexagonal.
This polyhedron is distinguished by a variety of properties:
- All parties and angles of the base are equal to each other;
- All edges and dihedral coal pyramids are also equal to each other;
- The triangles forming the sides are the same, respectively, they have the same area, side and height.
To calculate the area of \u200b\u200bthe correct hexagonal pyramid, the standard formula of the side surface area of \u200b\u200bthe hexagonal pyramid is applied:
where P is the perimeter of the base, A - the length of the apophem of the pyramid. In most cases, you can calculate the side area according to this formula, but sometimes you can use another method. Since the side faces of the pyramids are formed with equal triangles, you can find the area of \u200b\u200bone triangle, and then multiply it by the number of lateral sides. In the hexagonal pyramid of them 6. But this method can be applied and when calculating. Match an example of calculating the area of \u200b\u200bthe side surface of the hexagonal pyramid.
Let the correct hexagonal pyramid be given, in which the apophem is equal to a \u003d 7 cm, the base side B \u003d 3 cm. Calculate the area of \u200b\u200bthe lateral surface of the polyhedron.
To begin with, we will find the perimeter of the base. Since the pyramid is correct - in its foundation there is a regular hexagon. It means that all of its parties are equal, and the perimeter is calculated by the formula:
We substitute the data in the formula:
Now we can easily find the side surface area, substituting the value in the basic formula: 
Also an important point is to find the area of \u200b\u200bthe base. The formula of the base of the hexagonal pyramid is derived from the properties of the correct hexagon: 
Consider an example of calculating the area of \u200b\u200bthe base of the hexagonal pyramid, taking as the basis of the condition from the past example. Of course, we know that the base side B \u003d 3 cm. Substitute the data in the formula: 
The formula of the hexagonal pyramid is the sum of the base area and side sweep: 
Consider an example of calculating the hexagonal pyramid area.
Let the pyramid be given, at the base of which the correct hexagon with a side B \u003d 4 cm. Apofhem of a given polyhedron is equal to a \u003d 6 cm. Find the total area.
We know that the total area consists of the areas of the base and the side sweep. Therefore, to begin to find them. Calculate perimeter:
Now we find the side surface area: 
Next, we calculate the base area in which the right hexagon lies: 
Now we can fold the resulting results:
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