Volume of a regular 6-carbon pyramid. Volume of a hexagonal pyramid
Hexagonal pyramid called a polyhedron, the base of which is a regular hexagon, and the side faces are formed by identical isosceles triangles.
Such pyramids have many unique properties:
- All sides of the base are the same length;
- All side ribs are equal to each other;
- All angles at the base are equal, and also the dihedral angles formed by the edges are equal;
- Each side face is the same area.
It is calculated from the area of its base and lateral development. To calculate the volume, it is enough to know the height of the pyramid and the area of its base. First, let's look at the formula for the area of a regular hexagon.
One of the most significant differences between a regular hexagon and other figures is that its side is equal to the radius of the circumscribed circle. Thanks to this property, the base area of a regular hexagonal pyramid is calculated by the formula: 
For calculations, you can use both the radius of the circumscribed circle and the length of the side of a regular hexagon.
Now let's return to the formula for the volume of a hexagonal pyramid. It represents one third of the product of the area of the base and the height of the pyramid lowered to this base:
Now let's look at an example of calculating the volume of a hexagonal pyramid.
Let a regular hexagonal pyramid be given, the height of which is h = 8 cm. A circle with radius R = 6 cm is circumscribed around the base. Find the volume.
There will be nothing complicated in calculating the required parameter - after all, all the necessary quantities are specified by the conditions. Therefore, let's find the area of the base of our polyhedron. We remember that the radius of a circle circumscribed around a regular hexagon is equal to its sides. Let's substitute the data into the formula:
Now we can use the found area to calculate the volume of our hexagonal pyramid: 
In this way, knowing the properties of a regular hexagon and the volume formula for a hexagonal pyramid, we found all the necessary parameters.
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A pyramid whose base is a regular hexagon and whose sides are formed by regular triangles is called hexagonal.
This polyhedron has many properties:
- All sides and angles of the base are equal to each other;
- All edges and dihedral coals of the pyramid are also equal to each other;
- The triangles forming the sides are the same, respectively, they have the same areas, sides and heights.
To calculate the area of a regular hexagonal pyramid, the standard formula for the lateral surface area of a hexagonal pyramid is used:
where P is the perimeter of the base, a is the length of the apothem of the pyramid. In most cases, you can calculate the lateral area using this formula, but sometimes you can use another method. Since the side faces of the pyramid are formed by equal triangles, you can find the area of one triangle and then multiply it by the number of sides. There are 6 of them in a hexagonal pyramid. But this method can also be used when calculating. Let's consider an example of calculating the lateral surface area of a hexagonal pyramid.
Let a regular hexagonal pyramid be given, in which the apothem is a = 7 cm, the side of the base is b = 3 cm. Calculate the area of the lateral surface of the polyhedron.
First, let's find the perimeter of the base. Since the pyramid is regular, there is a regular hexagon at its base. This means that all its sides are equal, and the perimeter is calculated by the formula:
Substitute the data into the formula:
Now we can easily find the lateral surface area by substituting the found value into the basic formula: 
Also important is the search for the base area. The formula for the area of the base of a hexagonal pyramid is derived from the properties of a regular hexagon: 
Let's consider an example of calculating the area of the base of a hexagonal pyramid, taking as a basis the conditions from the previous example. From them we know that the side of the base b = 3 cm. Substitute the data into the formula: 
The formula for the area of a hexagonal pyramid is the sum of the area of the base and the side scan: 
Let's consider an example of calculating the area of a hexagonal pyramid.
Let a pyramid be given at the base of which lies a regular hexagon with side b = 4 cm. The apothem of the given polyhedron is a = 6 cm. Find the total area.
We know that the total area consists of the base and side scan areas. So let's find them first. Let's calculate the perimeter:
Now let's find the lateral surface area: 
Next, we calculate the area of the base in which the regular hexagon lies: 
Now we can add up the results: