Everyone has heard of the great Egyptian stone structures, the main of which is the pyramid of Cheops. In the course of stereometry, the characteristics of various pyramids are considered. One of the important parameters of the figure is the lateral surface area. According to what formula of the lateral surface the area of the pyramid should be calculated, this article will tell.
What is a pyramid in geometry?
Before talking about the pyramid and the formula for the area of the lateral surface, we give a definition of the figure itself. Under it is supposed a volumetric polyhedron consisting of one n-coal base and n triangles. All triangles have one side common with the base, and also intersect at a point called a vertex. The following is an arbitrary quadrangular pyramid:
Getting a pyramid is quite simple. To do this, select a flat polygon and connect all its vertices with a single point in space. A prerequisite is that this point should not lie on a plane.
Any pyramid consists of:
- faces that have n + 1 pieces;
- vertices (n + 1 piece);
- ribs (2 * n pieces).
Moreover, all these elements are of two types: those that belong to the base, and those that belong to the side surface.
Side surface options for an arbitrary type of shape
How to find the area (formula below) of the surface of the side face of the figure in question? It is not difficult to answer this question if you know that the side surface is formed by n triangles. This means that it is enough for each of them to calculate the area, and then add the obtained values and the result will be the desired indicator. However, this is not always easy for a pyramid of any type. We give an example. The figure below shows three pyramids called quadrangular sloping.
At first glance it is clear that all the side triangles are different. This means that to determine their area, you need to know all sides of the base and the height of each triangle. It is called an "apothem." If the apothem of the i-th triangle is denoted by h i , and the length of the corresponding side of the base is called a i , then for the general type of pyramid we obtain the formula for the lateral surface area:
S = 1/2 * ∑ i = 1 n (h i * a i ).
Thus, to calculate the value S of a figure of an arbitrary type, it is necessary to know 2 * n of its parameters.
Regular pyramids and their side surface
The surface area formula of the general pyramid given in the previous paragraph takes a concrete form for the correct figures. Correct is the pyramid that contains an equilateral and equiangular figure at the base, and its height falls exactly in the center of the base. The figure below shows a set of regular pyramids made of paper:
The fact that all triangles of the lateral surface are isosceles and are equal to each other for the correct pyramid, greatly facilitates the calculation of the surface area of its sidewall. The length of the side of the base is denoted by the letter a, and the apothem is denoted by h 1 , then for the pyramid the formula for the lateral surface area will take the form:
S = 1/2 * n * a * h 1 .
It is important not to confuse the value of h 1 in the formula with the height h of the pyramid. Apothem h 1 and height h are connected by a single equality through the length of the base for any regular pyramid.
The task of calculating the lateral surface of a triangular pyramid
It is known that a triangular regular pyramid has a height of 43 cm and a base length of 12 cm. What is the area of its lateral surface?
Having examined the right triangle inside this pyramid, which is formed by the sides h 1 , h and 1/3 of the height of the base, we get:
h 1 = √ (h 2 + a 2/12) = √ (43 2 +12 2/12) = 43.14 cm.
Now it remains to apply the above formula for S, taking into account that n = 3. We get:
S = 1/2 * n * a * h 1 = 1/2 * 3 * 12 * 43.14 = 776.52 cm 2 .
The written formula for determining apothems through height is valid only for a triangular regular pyramid.