The quadrangular pyramid is perhaps the most famous figure from this class of volumetric geometric objects. Its properties and characteristics are studied in high school. This article is intended to answer the question of what formula calculates the area of a quadrangular pyramid.
Quadrangular Pyramid
In order not to go far for examples of this figure, we immediately say that the great pyramid of Cheops is the most famous quadrangular regular figure.
From a purely geometric point of view, a quadrangular pyramid is an object formed by five faces: four triangles and one flat quadrangle. To build in space this figure is not difficult. To do this, take a flat quadrangle (square, rectangle, rhombus, parallelogram, and so on), and then all its vertices are connected to one single point in space, which will become the top of the pyramid. As a result of such simple geometric operations, we get a quadrangular pyramid.
It can be seen that the figure consists of five faces, five vertices, one of which is the main one, and eight edges (4 belong to the base, 4 belong to triangles).
Not all quadrangular pyramids have the same shape. There are several types of these shapes. For example, pyramids are inclined and straight. In the first case, the perpendicular, which is lowered from the top to the quadrangular base, intersects the latter at a point that does not coincide with its center. In the case of a straight figure, the intersection point of the perpendicular to the base plane is its center. Recall that the center of a convex quadrilateral lies at the intersection of two diagonals.
In addition to inclined and straight figures, quadrangular pyramids can be right and wrong. Any square base pyramid that is straight will be correct. Regular pyramids differ from each other in size (length of the side of the square a, length of the edges of the side b and height h). When performing calculations of various geometric characteristics with regular pyramids, in view of their high symmetry, it is convenient to work. In addition, many properties of these figures are described by special expressions, including the area formula of a regular quadrangular pyramid.
Area of a pyramid with a quadrangular base of arbitrary type
To determine the area of any polyhedron, it is necessary to add the area of all its sides. The studied figure has five sides, four of which are triangular. It is not difficult to find their area if you know the height of each triangle h bi (it is the apotheme of the pyramid) and the length of each side of the quadrangle a i . Then, for the quadrangular pyramid, the formula for the lateral surface area will take the form:
S b = 1/2 * ∑ i = 1 4 (a i * h bi )
To the value of S b should be added the area of the quadrangle S 4 to obtain the area of the full surface of the pyramid. The value of S 4 is easy to determine if the sides a i and the angles of the quadrangle are known.
Area of the correct figure
As mentioned above, for a regular pyramid, the quadrangular surface area formula has a specific form. Get her.
Let's start by looking at the area of the base. Since it is an ordinary square, its area is calculated using a simple expression:
S 4 = a 2
Now pay attention to the side surface. It is represented by four identical triangles, which are also isosceles, or equilateral. All apothems of triangles are equal, we denote their length h b . The surface area of the side will be equal to:
S b = 2 * h b * a
Then the formula of the surface area of the quadrangular pyramid correct will take the following form:
S = S 4 + S b = a 2 + 2 * h b * a
Geometry solution
It is known that the edge of a regular pyramid, which has a square in the base, is equal to the length of the diagonal of this base. Knowing that the side of the square is 8 cm, it is necessary to determine the area of all the faces of this figure.
Since the diagonal of the square d is equal to the length of the edge of the side b, we get:
b = d = a * √2
Now it should be seen that in the studied pyramid, the edge b, apothem h b and half of the side of the square form a triangle with an angle of 90 o . This fact allows us to use the Pythagorean theorem to determine h b :
h b = √ (b 2 - a 2/4) = √ (2 * a 2 - a 2/4) = a * √7 / 2
Now you can apply the area formula of the quadrangular pyramid:
S = a 2 + 2 * h b * a = a 2 + 2 * a * √7 / 2 * a = a 2 * (1 + √7)
It remains to substitute the value of the side of the square from the condition and write down the answer: S = 233.33 cm 2 .