Volume of the triangular pyramid. Pyramid
A pyramid is a polyhedron with a polygon at its base. All faces, in turn, form triangles that converge at one vertex. Pyramids are triangular, quadrangular, and so on. In order to determine which pyramid is in front of you, it is enough to count the number of corners at its base. The definition of "pyramid height" is very common in geometry problems in school curriculum... In the article we will try to consider different ways finding it.
Parts of the pyramid
Each pyramid consists of the following elements:
- side facesthat have three corners and converge at the top;
- apothem is the height that descends from its top;
- the top of the pyramid is a point that connects the side edges, but does not lie in the plane of the base;
- base is a polygon that does not have a vertex;
- the height of the pyramid is a segment that intersects the top of the pyramid and forms a right angle with its base.
How to find the height of a pyramid if its volume is known
Through the formula V \u003d (S * h) / 3 (in the formula V is the volume, S is the base area, h is the height of the pyramid), we find that h \u003d (3 * V) / S. To consolidate the material, let's solve the problem right away. The triangular base is 50 cm 2, while its volume is 125 cm 3. Unknown height triangular pyramid, which we need to find. Everything is simple here: we insert data into our formula. We get h \u003d (3 * 125) / 50 \u003d 7.5 cm.
How to find the height of a pyramid if you know the length of the diagonal and its edges
As we remember, the height of the pyramid forms a right angle with its base. This means that the height, edge and half of the diagonal together form. Many, of course, remember the Pythagorean theorem. Knowing two dimensions, it will not be difficult to find the third quantity. Recall the well-known theorem a² \u003d b² + c², where a is the hypotenuse, and in our case the edge of the pyramid; b - the first leg or half of the diagonal and c - respectively, the second leg, or the height of the pyramid. From this formula c² \u003d a² - b².
Now the problem: in a regular pyramid, the diagonal is 20 cm, while the length of the rib is 30 cm. You need to find the height. We solve: c² \u003d 30² - 20² \u003d 900-400 \u003d 500. Hence c \u003d √ 500 \u003d about 22.4.
How to find the height of a truncated pyramid
It is a polygon that has a section parallel to its base. The height of a truncated pyramid is a line segment that connects its two bases. The height can be found at correct pyramidif the lengths of the diagonals of both bases are known, as well as the edge of the pyramid. Let the diagonal of the larger base be d1, while the diagonal of the smaller base is d2, and the edge has length l. To find the height, you can lower the heights from the two upper opposite points of the diagram to its base. We see that we have got two right triangle, it remains to find the lengths of their legs. To do this, subtract the smaller one from the larger diagonal and divide by 2. So we find one leg: a \u003d (d1-d2) / 2. After that, according to the Pythagorean theorem, we only have to find the second leg, which is the height of the pyramid.
Now let's look at the whole thing in practice. We have a task before us. The truncated pyramid has a square at the base, the diagonal length of the larger base is 10 cm, while the smaller one is 6 cm, and the edge is 4 cm. It is required to find the height. To begin with, we find one leg: a \u003d (10-6) / 2 \u003d 2 cm. One leg is 2 cm, and the hypotenuse is 4 cm.It turns out that the second leg or height will be 16-4 \u003d 12, that is, h \u003d √12 \u003d about 3.5 cm.
The main characteristic of any geometric shape in space is its volume. In this article, we will consider what a pyramid with a triangle at the base is, and also show how to find the volume of a triangular pyramid - regular full and truncated.
What is a triangular pyramid?
Everyone has heard of the ancient Egyptian pyramids, however, they are rectangular, regular, not triangular. Let's explain how to get a triangular pyramid.
Take an arbitrary triangle and connect all its vertices with some one point located outside the plane of this triangle. The formed figure will be called a triangular pyramid. It is shown in the figure below.
As you can see, the figure under consideration is formed by four triangles, which are generally different. Each triangle is a side or face of a pyramid. This pyramid is often called a tetrahedron, that is, a four-sided volumetric figure.
In addition to the sides, the pyramid also has edges (it has 6) and vertices (there are 4 of them).
triangular base
A figure obtained using an arbitrary triangle and a point in space will generally be an irregular inclined pyramid. Now imagine that the original triangle has the same sides, and the point in space is located exactly above its geometric center at a distance h from the plane of the triangle. The pyramid built using this initial data will be correct.
Obviously, the number of edges, sides and vertices for a regular triangular pyramid will be the same as for a pyramid built from an arbitrary triangle.
However, the correct figure has some distinctive features:
- its height, drawn from the top, will exactly intersect the base in the geometric center (the point of intersection of the medians);
- side surface such a pyramid is formed by three identical triangles that are isosceles or equilateral.
A regular triangular pyramid is not only a purely theoretical geometric object. Some structures in nature have its form, for example, the crystal lattice of diamond, where a carbon atom is connected to four of the same atoms covalent bonds, or a methane molecule, where the tops of the pyramid are formed by hydrogen atoms.
triangular pyramid
You can determine the volume of absolutely any pyramid with an arbitrary n-gon at the base using the following expression:
Here the symbol S o denotes the area of \u200b\u200bthe base, h is the height of the figure drawn to the marked base from the top of the pyramid.
Since the area of \u200b\u200ban arbitrary triangle is equal to half the product of the length of its side a by the apothem h a, dropped to this side, the formula for the volume of a triangular pyramid can be written in the following form:
V \u003d 1/6 × a × h a × h
For a general type, determining the height is not an easy task. To solve it, the easiest way is to use the formula for the distance between a point (vertex) and a plane (triangular base), represented by the equation general view.
For the correct one it has a specific look. The area of \u200b\u200bthe base (equilateral triangle) for it is:
We substitute it into the general expression for V, we get:
V \u003d √3 / 12 × a 2 × h
A special case is the situation when all sides of a tetrahedron turn out to be the same equilateral triangles. In this case, its volume can be determined only based on the knowledge of the parameter of its edge a. The corresponding expression is:
Truncated pyramid
If the upper part containing the vertex is cut off at a regular triangular pyramid, then you get a truncated figure. Unlike the original, it will consist of two equilateral triangular bases and three isosceles trapezoids.
The photo below shows what a regular truncated triangular pyramid made of paper looks like.
To determine the volume of a truncated triangular pyramid, you need to know its three linear characteristics: each of the sides of the bases and the height of the figure, equal to the distance between the upper and lower bases. The corresponding formula for volume is written as follows:
V \u003d √3 / 12 × h × (A 2 + a 2 + A × a)
Here h is the height of the figure, A and a are the side lengths of the large (lower) and small (upper) equilateral triangles, respectively.
The solution of the problem
To make the information provided in the article clearer for the reader, we will show with an illustrative example how to use some of the written formulas.
Let the volume of the triangular pyramid be 15 cm 3. The figure is known to be correct. The apothem a b of the lateral rib should be found if the height of the pyramid is known to be 4 cm.
Since the volume and height of the figure are known, you can use the appropriate formula to calculate the length of the side of its base. We have:
V \u003d √3 / 12 × a 2 × h \u003d\u003e
a \u003d 12 × V / (√3 × h) \u003d 12 × 15 / (√3 × 4) \u003d 25.98 cm
a b \u003d √ (h 2 + a 2/12) \u003d √ (16 + 25.98 2/12) \u003d 8.5 cm
The calculated length of the apothem of the figure turned out to be greater than its height, which is true for any type of pyramid.
Quadrangular pyramid is called a polyhedron, at the base of which is a square, and all side faces are the same isosceles triangles.
This polyhedron has many different properties:
- Its lateral edges and the adjacent dihedral angles are equal to each other;
- The areas of the side faces are the same;
- At the base of the correct quadrangular pyramid lies a square;
- The height dropped from the top of the pyramid intersects with the intersection of the base diagonals.
All of these properties make it easy to find. However, quite often, in addition to it, it is required to calculate the volume of the polyhedron. For this, the formula for the volume of a quadrangular pyramid is applied:
That is, the volume of the pyramid is equal to one third of the product of the height of the pyramid by the area of \u200b\u200bthe base. Since it is equal to the product of its equal sides, then we immediately enter the square area formula into the volume expression.
Let's consider an example of calculating the volume of a quadrangular pyramid.
Let a quadrangular pyramid be given, at the base of which lies a square with a side a \u003d 6 cm. The side face of the pyramid is b \u003d 8 cm. Find the volume of the pyramid. 
To find the volume of a given polyhedron, we need the length of its height. Therefore, we will find it by applying the Pythagorean theorem. First, let's calculate the length of the diagonal. In the blue triangle, it will be the hypotenuse. It is also worth remembering that the diagonals of the square are equal to each other and are divided in half at the point of intersection: 
Now from the red triangle we find the height h we need. It will be equal to: 
Substitute the required values \u200b\u200band find the height of the pyramid:
Now, knowing the height, we can substitute all values \u200b\u200bin the formula for the volume of the pyramid and calculate the required value:
In this way, knowing a few simple formulas, we were able to calculate the volume of a regular quadrangular pyramid. Do not forget that given value measured in cubic units.
One of the simplest volumetric figures is a triangular pyramid, since it consists of the smallest number of faces from which a figure in space can be formed. In this article, we will consider the formulas with which you can find the volume of a triangular regular pyramid.
Triangular pyramid
According to the general definition, a pyramid is a polygon, all vertices of which are connected to one point that is not located in the plane of this polygon. If the latter is a triangle, then the whole figure is called a triangular pyramid.
The pyramid in question consists of a base (triangle) and three side faces (triangles). The point where the three side faces are connected is called the top of the shape. The perpendicular dropped to the base from this top is the height of the pyramid. If the point of intersection of the perpendicular with the base coincides with the point of intersection of the medians of the triangle at the base, then they speak of a regular pyramid. Otherwise, it will be oblique.
As mentioned, the base of a triangular pyramid can be a general triangle. However, if it is equilateral, and the pyramid itself is straight, then they talk about the correct volumetric figure.
Any triangular pyramid has 4 faces, 6 edges, and 4 vertices. If the lengths of all edges are equal, then such a figure is called a tetrahedron.
general type
Before writing down a regular triangular pyramid, we give an expression for this physical quantity for a general pyramid. This expression looks like:
Here S o - base area, h - figure height. This equality will be true for any type of base of a pyramid polygon, as well as for a cone. If at the base there is a triangle with side length a and height h o lowered onto it, then the formula for the volume will be written as follows:
Volume formulas for a regular triangular pyramid
A regular triangular pyramid has an equilateral triangle at its base. It is known that the height of this triangle is related to the length of its side by equality:
Substituting this expression into the formula for the volume of a triangular pyramid written in the previous paragraph, we get:
V \u003d 1/6 * a * h o * h \u003d √3 / 12 * a 2 * h.
The volume of a regular pyramid with a triangular base is a function of the length of the base side and the height of the figure.
Since anyone regular polygon can be inscribed in a circle, the radius of which will uniquely determine the length of the side of the polygon, then this formula can be written through the corresponding radius r:
This formula can be easily obtained from the previous one, if we take into account that the radius r of the circumscribed circle through the length of the side a of the triangle is determined by the expression:
The problem of determining the volume of a tetrahedron
Let us show how to use the above formulas in solving specific problems of geometry.
It is known that a tetrahedron has an edge length of 7 cm. Find the volume of a regular triangular pyramid-tetrahedron.
Recall that the tetrahedron is regular in which all bases are equal. To use the triangular volume formula, you need to calculate two quantities:
- the length of the side of the triangle;
- figure height.
The first value is known from the problem statement:
To determine the height, consider the figure shown in the figure.
The marked triangle ABC is right-angled, where the angle ABC is 90 o. The AC side is the hypotenuse, the length of which is a. By simple geometric reasoning, one can show that the side BC has the length:
Note that the length BC is the radius of a circle circumscribed around a triangle.
h \u003d AB \u003d √ (AC 2 - BC 2) \u003d √ (a 2 - a 2/3) \u003d a * √ (2/3).
Now we can substitute h and a in the corresponding formula for volume:
V \u003d √3 / 12 * a 2 * a * √ (2/3) \u003d √2 / 12 * a 3.
Thus, we have obtained the formula for the volume of a tetrahedron. It can be seen that the volume depends only on the length of the rib. If we substitute the value from the condition of the problem into the expression, then we get the answer:
V \u003d √2 / 12 * 7 3 ≈ 40.42 cm 3.
If we compare this value with the volume of a cube having the same edge, we get that the volume of a tetrahedron is 8.5 times less. This indicates that the tetrahedron is a compact figure that is realized in some natural substances. For example, a methane molecule is tetrahedral, and each carbon atom in a diamond is attached to four other atoms to form a tetrahedron.
The problem with homothetic pyramids
Let's solve an interesting geometric problem. Suppose there is a triangular regular pyramid with some volume V 1. How many times should the size of this figure be reduced in order to obtain a pyramid homothetic to it with a volume three times smaller than the initial one?
Let's start solving the problem by writing the formula for the original regular pyramid:
V 1 \u003d √3 / 12 * a 1 2 * h 1.
Let the volume of the figure, necessary according to the condition of the problem, be obtained if we multiply its parameters by the coefficient k. We have:
V 2 \u003d √3 / 12 * k 2 * a 1 2 * k * h 1 \u003d k 3 * V 1.
Since the ratio of the volumes of figures is known from the condition, we obtain the value of the coefficient k:
k \u003d ∛ (V 2 / V 1) \u003d ∛ (1/3) ≈ 0.693.
Note that we would get a similar value for the coefficient k for an arbitrary type of pyramid, and not just for a regular triangular one.
