Stereometry, as a section of geometry in space, studies the properties of prisms, cylinders, cones, balls, pyramids and other three-dimensional figures. This article is devoted to a detailed discussion of the characteristics and properties of a hexagonal regular pyramid.
What pyramid will be studied
The regular hexagonal pyramid is a figure in space, which is limited by one equilateral and equiangular hexagon, and six identical triangles isosceles. These triangles can also be equilateral under certain conditions. This pyramid is shown below.
The same figure is depicted here, only in one case is it turned by a side face towards the reader, and in the other by a side edge.
The regular hexagonal pyramid has 7 faces, which were named above. She also owns 7 peaks and 12 edges. Unlike prisms, all pyramids have one particular vertex, which is formed by the intersection of side triangles. For the correct pyramid, it plays an important role, since the perpendicular is lowered from it to the base of the figure. Further, the height will be denoted by the letter h.
The pyramid shown is called correct for two reasons:
- at its base is a hexagon with equal lengths of sides a and with equal angles of 120 o ;
- the height of the pyramid h intersects the hexagon exactly in its center (the intersection point lies at the same distance from all sides and from all vertices of the hexagon).
Surface area
We begin to consider the properties of a regular hexagonal pyramid by determining its area. To do this, it is first useful to bring the scan pattern on a plane. Its schematic image is shown below.
It can be seen that the sweep area, and hence the entire surface of the figure in question, is equal to the sum of the areas of six identical triangles and one hexagon.
To determine the area of the hexagon S 6, we use the universal formula for a regular n-gon:
S n = n / 4 * a 2 * ctg (pi / n) =>
S 6 = 3 * √3 / 2 * a 2 .
Where the letter a denotes the length of the side of the hexagon.
The area of the triangle S 3 of the side can be found if you know the value of its height h b :
S 3 = 1/2 * h b * a.
Since all six triangles are equal, we obtain a working expression for determining the area of a hexagonal pyramid with the correct base:
S = S 6 + 6 * S 3 = 3 * √3 / 2 * a 2 + 6 * 1/2 * h b * a = 3 * a * (√3 / 2 * a + h b ).
Pyramid volume
Just like the area, the volume of the hexagonal regular pyramid is an important property of it. This volume is calculated by the general formula for all pyramids and cones. We write it down:
V = 1/3 * S o * h.
Here, the symbol S o is the area of the hexagonal base, that is, S o = S 6 .
Substituting the above expression for S 6 into the formula for V, we arrive at a finite equality for determining the volume of the correct hexagonal pyramid:
V = √3 / 2 * a 2 * h.
An example of a geometric problem
In a hexagonal pyramid, the regular side rib is twice the length of the side of the base. Knowing that the latter is 7 cm, it is necessary to calculate the surface area and volume of this figure.
As you might guess, the solution to this problem involves the use of the above expressions for S and V. Nevertheless, they cannot be used immediately, since we do not know the apothem and height of the regular hexagonal pyramid. Let's deal with their calculation.
The apothem h b can be determined by looking at the right triangle, built on the sides b, a / 2 and h b . Here b is the length of the side rib. Using the condition of the problem, we obtain:
h b = √ (b 2 -a 2/4) = √ (14 2 -7 2/4) = 13.555 cm.
The height h of the pyramid can be determined in exactly the same way as the apothem, only now we should consider a triangle with sides h, b and a located inside the pyramid. The height will be equal to:
h = √ (b 2 - a 2 ) = √ (14 2 - 7 2 ) = 12.124 cm.
It can be seen that the calculated height is less than that for the apothem, which is true for any pyramid.
Now you can use the expressions for volume and area:
S = 3 * a * (√3 / 2 * a + h b ) = 3 * 7 * (√3 / 2 * 7 + 13.555) = 411.96 cm 2 ;
V = √3 / 2 * a 2 * h = √3 / 2 * 7 2 * 12.124 = 514.48 cm 3 .
Thus, to uniquely determine any characteristic of a regular hexagonal pyramid, it is necessary to know any two of its linear parameters.