When studying the properties of figures in three-dimensional space in the framework of stereometry, it is often necessary to solve problems of determining the volume and surface area. In this article, we show how to calculate the volume and area of the side surface for a truncated pyramid using well-known formulas.
Pyramid in geometry
In geometry, an ordinary pyramid is a shape in space that is built on some flat n-gon. All its vertices are connected to one point located outside the plane of the polygon. For an example we give a photo where the pentagonal pyramid is represented.
This figure is formed by faces, vertices and edges. The pentagonal face is called the base. The remaining triangular faces form a side surface. The intersection point of all triangles is the main peak of the pyramid. If from it to lower the perpendicular to the base, then two options for the position of the intersection point are possible:
- in the geometric center, then the pyramid is called a straight line;
- not in the geometric center, then the figure will be inclined.
Further we will consider only straight figures with the correct n-coal base.
What kind of figure is this - a truncated pyramid?
To determine the volume of a truncated pyramid, it is necessary to clearly understand what kind of figure is in question. We will clarify this issue.
Suppose we took a secant plane that is parallel to the base of a regular pyramid, and cut off part of the side surface with it. If you do this operation with the pentagonal pyramid shown above, you get a figure like the one below.
It can be seen from the photo that this pyramid already has two bases, and the upper one is similar to the lower one, but it is smaller in size. The lateral surface is no longer represented by triangles, but by trapezoids. They are isosceles, and their number corresponds to the number of sides of the base. A truncated figure does not have a main peak, like a regular pyramid, and its height is determined by the distance between parallel bases.
In the general case, if the figure in question is formed by n-carbon bases, it has n + 2 faces, or sides, 2 * n vertices and 3 * n edges. That is, a truncated pyramid is a polyhedron.
The formula for the volume of a truncated pyramid
Recall that the volume of an ordinary pyramid is 1/3 of the product of its height and base area. For a truncated pyramid, this formula is not suitable, since it has two bases. And its volume will always be less than the same value for the ordinary figure from which it is derived.
Without going into the mathematical details of obtaining the expression, we give the final formula for the volume of the truncated pyramid. It is written as follows:
V = 1/3 * h * (S 1 + S 2 + √ (S 1 * S 2 ))
Here S 1 and S 2 are the areas of the lower and upper bases, respectively, h is the height of the figure. The written expression is true not only for the direct regular truncated pyramid, but also for any figure of this class. Moreover, regardless of the type of base polygons. The only condition limiting the use of the expression for V is the necessity of parallelism of the pyramid bases.
Several important conclusions can be drawn by studying the properties of this formula. So, if the area of the upper base is zero, then we come to the formula for V of the ordinary pyramid. If the area of the bases are equal to each other, then we obtain the formula for the volume of the prism.
How to determine the lateral surface area?
Knowing the characteristics of a truncated pyramid involves not only the ability to calculate its volume, but also know how to determine the area of the side surface.
A truncated pyramid consists of two types of faces:
- isosceles trapezoid;
- polygonal bases.
If there is a regular polygon in the bases, then calculating its area is not very difficult. To do this, you only need to know the length of the side a and their number n.
In the case of a lateral surface, the calculation of its area involves the determination of this value for each of n trapezoids. If the n-gon is correct, then the formula for the lateral surface area takes the form:
S b = h b * n * (a 1 + a 2 ) / 2
Here h b is the height of the trapezoid, which is called the apothem of the figure. The values of a 1 and a 2 are the lengths of the sides of regular n-coal bases.
For each regular n-angular truncated pyramid, we can uniquely determine the apotheme h b through the parameters a 1 and a 2 and the height h of the figure.
The task of calculating the volume and area of the figure
Given a regular triangular truncated pyramid. It is known that its height h is 10 cm, and the lengths of the sides of the bases are 5 cm and 3 cm. What is the volume of the truncated pyramid and the area of its lateral surface?
First, we calculate the value V. To do this, find the area of equilateral triangles located at the base of the figure. We have:
S 1 = √3 / 4 * a 1 2 = √3 / 4 * 5 2 = 10.825 cm 2 ;
S 2 = √3 / 4 * a 2 2 = √3 / 4 * 3 2 = 3.897 cm 2
Substitute the data in the formula for V, we obtain the desired volume:
V = 1/3 * 10 * (10.825 + 3.897 + √ (10.825 * 3.897)) ≈ 70.72 cm 3
To determine the lateral surface, one should know the length of the apotheme h b . Considering the corresponding right-angled triangle inside the pyramid, we can write for it the equality:
h b = √ ((√3 / 6 * (a 1 - a 2 )) 2 + h 2 ) ≈ 10.017 cm
The value of the apotheme and sides of the triangular bases is substituted into the expression for S b and we get the answer:
S b = h b * n * (a 1 + a 2 ) / 2 = 10.017 * 3 * (5 + 3) / 2 ≈ 120.2 cm 2
Thus, we answered all the questions of the problem: V ≈ 70.72 cm 3 , S b ≈ 120.2 cm 2 .