In this article, we will consider how to calculate the area of a regular pyramid. The geometric properties of figures in space are studied by stereometry. This section of geometry uses the concepts of line lengths and the angles between them to define such characteristics of shapes as area or volume.
What kind of figure is this?
Before proceeding to consider specific formulas for the area of a regular pyramid (lateral surface and base), it should be clarified what it represents in terms of geometry. Let’s explain.
A pyramid is any geometric object in space that is formed as a result of connecting one point with all the vertices of a certain polygon. So the figure is formed. A common point is called a vertex and the plane of the polygon does not belong.
Three different pyramids are shown above. It can be seen that they differ in the number of sides of the polygonal base (three, four and five). However, all these figures are united by the fact that their lateral surface consists of triangles. Their number is different, it is equal to the number of sides of the base.
It can be seen from the figure that any pyramid consists of edges, vertices and faces. Their combination forms a single object. If from its top (the intersection point of the side triangles) to lower the perpendicular to the base, then its length will be the height of the figure. If this segment ends in the center of the base, then such a pyramid will be called a straight line. We figured it out, go ahead.
A straight pyramid whose base is an equilateral and equiangular polygon is called regular. We give an example. The Cheops pyramid is a regular quadrangular figure, because it is straight and has a square base.
Value Formulas
In stereometry, their sweeps are used to study the area of the regular pyramid and other three-dimensional figures. Explain graphically. The figure shows what a scan of a regular quadrangular pyramid is. Now it becomes more clear how to make calculations.
It can be seen that the area of the regular pyramid has two components: the base and the side surface. Assign variables.
Denote the area of the base by the symbol S o . To calculate this quantity, there is a universal formula for a regular n-gon. It has the form:
S n = n / 4 * a 2 * ctg (pi / n).
Here n is the number of sides of the base, a is the side of the n-gon.
In the regular pyramid, all the triangles that form the side surface are equal to each other and in the general case are isosceles. This means that in order to determine the shape of the side surface area (denoted by S b ), it is necessary to find this parameter in one triangle, and then multiply by their number. We write the formula. The area of the side of the regular pyramid is calculated as follows:
S b1 = 1/2 * a * h b.
Here h b is the height of the triangle, which is called the apothem of the pyramid.
Since the number of triangles is equal to the number of sides of the n-carbon base, we obtain the formula for calculating the value of S b :
S b = n * S b1 = n * a * h b / 2.
Then the total area of the regular pyramid can be calculated by the formula:
S = S o + S b = n / 4 * a 2 * ctg (pi / n) + n * a * h b / 2.
The solution of the problem
Condition. It is known that a regular quadrangular pyramid has a base side length of 10 cm and a height of 12 cm. What is its surface area?
Decision. The surface area of the base (square) is equal to:
S o = a 2 = 100 cm 2 .
To calculate the lateral surface area, it is necessary to know the apothem h b of the figure. If we consider a right triangle inside the pyramid, then the apothem, which is a hypotenuse in it, is calculated as follows:
h b = √ (a 2/4 + h 2 ) = 13 cm.
Then the area of the side surface of the regular pyramid is equal to:
S b = n * a * h b / 2 = 4 * 10 * 13/2 = 260 cm 2 .
Adding the obtained values of S o and S b , we come to the answer to the question of the problem: S = 360 cm 2 .
The article describes how to find the area of a figure called the regular pyramid.