One of the polyhedrons that are taught in schools in the course of spatial geometry is the pyramid. This figure has a number of parameters and characteristics, for the calculation of which certain mathematical formulas are used. After reading the article, you will learn how to find the surface area of ββa pyramid.
What is a pyramid? Types of figures
This is a figure in three-dimensional space, which is a polyhedron consisting of triangles and a polygon. If we take an arbitrary polygon on the plane and connect all its vertices with straight segments with some point that does not belong to the plane of this polygon, we get a pyramid of arbitrary type.
A pyramid consists of faces, vertices and edges. Faces are planes that limit the volume of a figure. The faces are separated by ribs. If three faces intersect at one point, then the latter is a vertex. Any such figure has several vertices, for example, a triangular figure has four, and a quadrangular has five. Each pyramid has only one vertex that does not belong to the base. It is called the main or main.
The pyramid class includes several types of shapes. A pyramid will be straight if its lateral triangles are isosceles. If these triangles are also equal to each other, then the figure will be correct. For any straight and regular pyramid, the height (the distance from the main peak to the base) intersects the base in its geometric center. In addition, the correct figure has an equilateral (equiangular) base.
Pyramid surface area
The area of ββany such figure is understood as the sum of the areas of all its sides. Since pyramids are of different types, there is no universal formula for calculating their areas. However, there are expressions that can be used in each case.
Whatever the pyramid, it always consists of an n-angular face and n triangles. The area of ββtriangles is easy to calculate if their heights and sides of the base are known. As for the n-gon, to determine its area it is necessary to analyze what kind of n-gon it is, whether it is correct, what angles are known. A universal method for determining its area is to divide it into simpler figures, for example, triangles or parallelograms.
Correct figure
For the correct pyramid, the surface area formula has long been determined. Before recording it, we note that the area of ββthe correct base of the figure can be calculated as follows:
S n = n / 4 * a 2 * ctg (pi / n).
In the formula: a is the side of the polygon, n is the number of sides in it. For example, for a triangle, the formula is as follows:
S 3 = β3 / 4 * a 2 .
For the square, we get a typical equality:
S 4 = a 2
For the correct pyramid, the surface area of ββthe side S b can be determined by the following formula:
S b = n / 2 * h b * a.
Where h b is the apothem of the pyramid (the height of the lateral triangle).
Adding the expressions for S n and S b , we obtain the formula for the area of ββthe full surface of the pyramid:
S = S n + S b = n / 4 * a 2 * ctg (pi / n) + n / 2 * h b * a.
Note that in order to uniquely determine S of any pyramid, it is necessary to know its two linear parameters.
Tilted figure
The surface area of ββan oblique pyramid is much more difficult to calculate than the correct one. Nevertheless, knowing its development, you can always solve the problem. The side surface of an inclined figure is calculated as follows:
S b = 1/2 * β i = 1 n (a i * h bi ).
Here a i is the length of the i-th side of the base, h bi is the length of the i-th apothem. Apothems for the inclined pyramid of the general type differ.
The base area S o is calculated based on its type, for example, if it is a parallelogram with sides a 1 and a 2 and the angle between them ΞΈ, then we can write:
S o = a 1 * a 2 * sin (ΞΈ).
For both the inclined and the straight pyramids, the apothems are associated with the lengths of the side ribs and the ribs of the base. This connection is often used in solving problems.