A pyramid is a three-dimensional figure, the base of which is a polygon, and the sides are triangles. The hexagonal pyramid is its private view. In addition, there are other variations when at the base of the triangle (such a figure is called a tetrahedron) there is a square, a rectangle, a pentagon, and so on in increasing order. When the number of points becomes infinite, a cone is obtained.
Hexagonal pyramid
In general, this is one of the last and most difficult topics in stereometry. It is studied somewhere in grades 10-11 and only the option is considered when the correct figure is at the base. One of the most difficult tasks in the USE is often associated with this paragraph.
And so, at the base of a regular hexagonal pyramid lies a regular hexagon. What does it mean? At the base, all sides are equal. The lateral parts consist of isosceles triangles. Their vertices touch at one point. This figure is presented in the photo below.
How to find the area of the entire surface and the volume of the hexagonal pyramid?
Unlike the mathematics taught at universities, school science teaches you to bypass and simplify some complex concepts. For example, if you don’t know how to find the area of a figure, you have to divide it into parts and find the answer using the well-known area formulas of the divided figures. This principle must be followed in the case presented.
That is, to find the surface area of the entire hexagonal pyramid, you need to find the base area, then the area of one of the sides and multiply it by 6.
The following formulas are used:
S (full) = 6S (side) + S (base), (1);
S (base) = 3√3 / 2a 2 , (2);
6S (side) = 6 × 1 / 2ab = 3ab, (3);
S (full) = 3ab + (3√3 / 2a 2 ) = 3 (2a 2b + √3) / 2a 2 , (4).
Where S is the area, cm 2 ;
a is the length of the base, cm;
b - apothem (height of the lateral face), see
In order to find the area of the entire surface or any of its components, it is required only the side of the base of the hexagonal pyramid and apothem. If the problem gives this condition, then the solution should not be difficult.
The volume of things is much easier, but to find it, you need the height (h) of the hexagonal pyramid itself. And, of course, the side of the foundation, thanks to which it is necessary to find its area.
The formula is as follows:
V = 1/3 × S (base) × h, (5).
Where V is the volume, cm 3 ;
h is the height of the figure, see
A variant of the problem that may fall on the exam
Condition. Given a regular hexagonal pyramid. The length of the base is 3 cm. The height is 5 cm. Find the volume of this figure.
Solution: V = 1/3 × (3√3 / 2 × 3 2 ) × 5 = 5/3 × √3 / 6 = 5√3 / 18.
Answer: the volume of the regular hexagonal pyramid is 5√3 / 18 cm.