Studying volumetric figures in the course of stereometry, schoolchildren often face the task of determining their surface area. A successful solution to this problem is possible if you clearly understand what kind of figure is being worked on. This article is devoted to the issue of determining the surface area of a regular pyramid (quadrangular).
Description of the figure
We begin the disclosure of the question of the article with the definition of a regular quadrangular pyramid. In geometry, it is understood as a figure in space, which is formed by one square and four identical isosceles triangles (with certain parameters of the figure, these triangles can be equilateral). An example of the figure in question is shown below.
Each isosceles triangle intersects along the side faces with two adjacent triangles, and along the edge of the base - with a square. In addition, all four triangles intersect at one point, which is called the main peak of the pyramid. In addition to her, the figure has four more vertices, but they all belong to the base.
This pyramid is called quadrangular because its base is a quadrangle. And it is considered correct because this base is a square, and the figure itself is straight. The latter means that the perpendicular dropped to the square intersects it exactly at the geometric center.
What is the surface area of a regular pyramid (quadrangular)
It will not be easy for many to answer this question. Indeed, we consider a three-dimensional figure, and when we talk about areas, we mean a silhouette on a plane. In connection with the above, in geometry, to calculate the areas of spatial objects, their flat sweeps are used.
The development of the pyramid in question is easy to obtain. Suppose we have a paper figure with a square base. Take scissors and cut a square from it. Then we cut along the side edge (the segment of the intersection of triangles) the pyramidal surface and expand it into a flat interpretation. As a result of these actions, we will get a flat figure, similar to the one shown in the figure.
Thus, answering the question of how to find the area of a regular quadrangular pyramid, it should be said that for this it is necessary to add the area of the square and the area of four identical triangles.
Formulas for finding the value
From planimetry it is known that the calculation of the area is made taking into account the knowledge of the linear parameters of a flat figure. In our case, we are talking about two types of objects: an isosceles triangle and a square.
Denote the side of the quadrangle by the letter a, and the height of the triangle h b (it is called the apothem of the pyramid). Then for the area S o square you can write:
S o = a 2
The area of the triangle S 3 will be equal to:
S 3 = 1/2 * a * h b
Since there are four pieces of triangles with area S 3 in the pyramid under consideration, the surface area formula of the regular pyramid (quadrangular) will take the form:
S = S o + 4 * S 3 = a 2 + 4/2 * a * h b = a * (a + 2 * h b )
When solving some problems, instead of the apotheme h b , another linear parameter of the pyramid can be known - the height h. Therefore, it will be useful if we present a formula for S in terms of the parameters a and h.
You can solve the problem if you see a triangle inside the pyramid and calculate the hypotenuse-apothem using the following formula:
h b = √ (h 2 + a 2/4)
Substituting this expression in the above formula for S, we get:
S = a * (a + 2 * √ (h 2 + a 2/4))
This expression looks a bit more complicated than the first. However, it is more often used when considering geometric problems with a quadrangular pyramid.
The written formulas for the area of the figure in question can not be remembered, it is only important to clearly understand the scan of the pyramid and be able to find the area of the triangle.
Using the area formula of a quadrangular pyramid as an example of the construction of Cheops
Of course, we are talking about the Pyramid of Cheops - the most famous stone building in the history we know. We calculate the surface area of this giant using the following data about it:
- the average length of the side of the base is 230.363 meters;
- the initial height of the structure was 14.60 meters.
To find the desired area, you should use the second formula given in the previous paragraph of the article. Let's do it:
S = a * (a +2 * √ (h 2 + a 2/4)) = 230.363 2 + 230.363 * 2 * √ (146.50 2 + 230.363 2/4) ≈ 138 927 m 2
To understand how huge the calculated value is, compare it with the parameters of a football field (5,000 m 2 ). The total surface area of the regular quadrangular pyramid of Cheops is almost 28 times the size of the playing area.