Quadrangular pyramid is a polyhedron whose base is a square, and all its side faces are identical isosceles triangles.
This polyhedron has many different properties:
- Its lateral edges and adjacent dihedral angles are equal to each other;
- The areas of the side faces are the same;
- At the base of a regular quadrangular pyramid lies a square;
- The height dropped from the top of the pyramid intersects the point where the diagonals of the base intersect.
All these properties make it easy to find. However, quite often, in addition to this, it is necessary to calculate the volume of the polyhedron. To do this, use the formula for the volume of a quadrangular pyramid:
That is, the volume of the pyramid is equal to one third of the product of the height of the pyramid and the area of the base. Since it is equal to the product of its equal sides, we immediately enter the formula for the area of a square into the expression for volume.
Let's consider an example of calculating the volume of a quadrangular pyramid.
Let a quadrangular pyramid be given, the base of which is a square with side a = 6 cm. The side face of the pyramid is b = 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 a 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: 
Let's substitute the necessary values and find the height of the pyramid:
Now, knowing the height, we can substitute all the values into 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. Remember that this value is measured in cubic units.
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A polyhedron whose base is a regular triangle and the remaining faces are represented by isosceles triangles is called triangular pyramid Such a pyramid is also called a tetrahedron.
A regular pyramid has many properties that are derived from its constituent figures:
- All sides of the base are equal to each other because it is represented by a regular triangle;
- All edges of the pyramid are also equal to each other;
- Because each face forms an isosceles triangle in which the edges are equal and the bases are equal, then we can say that the area of each face is the same;
- All dihedral angles at the base are equal.
It is calculated as the sum of the areas of the base and the side scan. It can also be found by calculating the area of one of the side faces and the base. The formula for the volume of a triangular pyramid is also derived from the properties of the triangles of which it consists:
The base area is calculated from the formula: 
Let's consider an example of calculating the volume of a triangular pyramid.
Let us be given a triangular pyramid. The side of the base is a = 2 cm, and the height is h = 2√3. Find the volume of the given polyhedron.
First, let's find the area of the base. To do this, let’s substitute the known data into the above formula: 
Now we use the found value to calculate the volume of a triangular pyramid: 
You can also use a shortened formula to calculate the area of a triangular pyramid. It combines the area of the base and the height, and the formula is read as a third of the product of the base area and the height of the pyramid:
When using this formula, it is important to strictly follow the calculations and reductions. One small mistake can lead to an incorrect result. In general, finding the volume of a regular triangular pyramid is very simple.
One of the simplest three-dimensional figures is the triangular pyramid, since it consists of the smallest number of faces from which a figure can be formed in space. In this article we will look at formulas that can be used to find the volume of a triangular regular pyramid.
Triangular pyramid
According to the general definition, a pyramid is a polygon, all of whose vertices are connected to one point that is not located in the plane of this polygon. If the latter is a triangle, then the entire figure is called a triangular pyramid.
The pyramid in question consists of a base (triangle) and three side faces (triangles). The point at which the three side faces are connected is called the vertex of the figure. The perpendicular from this vertex dropped to the base 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 we speak of a regular pyramid. Otherwise it will be slanted.
As stated, the base of a triangular pyramid can be a general type of triangle. However, if it is equilateral, and the pyramid itself is straight, then they speak of a regular three-dimensional 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 type pyramid. This expression looks like:
Here S o is the area of the base, h is the height of the figure. This equality will be valid for any type of pyramid polygon base, 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 volume will be written as follows:
Formulas for the volume of a regular triangular pyramid
A regular triangular pyramid has an equilateral triangle at the base. It is known that the height of this triangle is related to the length of its side by the equality:
Substituting this expression into the formula for the volume of a triangular pyramid written in the previous paragraph, we obtain:
V = 1/6*a*h o *h = √3/12*a 2 *h.
The volume of a regular pyramid with a triangular base is a function of the length of the side of the base and the height of the figure.
Since any 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 in terms of 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 side a of the triangle is determined by the expression:
Problem of determining the volume of a tetrahedron
We will show how to use the above formulas when solving specific geometry problems.
It is known that a tetrahedron has an edge length of 7 cm. Find the volume of a regular triangular pyramid-tetrahedron.
Recall that a tetrahedron is regular in which all bases are equal to each other. To use the triangular volume formula, you need to calculate two quantities:
- length of the side of the triangle;
- height of the figure.
The first quantity is known from the problem conditions:
To determine the height, consider the figure shown in the figure.
The marked triangle ABC is a right triangle, where angle ABC is 90 o. Side AC is the hypotenuse and its length is a. Using simple geometric reasoning, it can be shown that side BC has the length:
Note that the length BC is the radius of the circle circumscribed around the triangle.
h = AB = √(AC 2 - BC 2) = √(a 2 - a 2 /3) = a*√(2/3).
Now you can substitute h and a into the corresponding formula for volume:
V = √3/12*a 2 *a*√(2/3) = √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 edge. If we substitute the value from the problem conditions into the expression, then we get the answer:
V = √2/12*7 3 ≈ 40.42 cm 3.
If we compare this value with the volume of a cube having the same edge, we find that the volume of the tetrahedron is 8.5 times less. This indicates that the tetrahedron is a compact figure that occurs in some natural substances. For example, the methane molecule has a tetrahedral shape, and each carbon atom in diamond is connected to four other atoms to form a tetrahedron.
Homothetic pyramid problem
Let's solve one interesting geometric problem. Suppose that there is a triangular regular pyramid with a certain volume V 1. How many times should the size of this figure be reduced in order to obtain a homothetic pyramid with a volume three times smaller than the original?
Let's start solving the problem by writing the formula for the original regular pyramid:
V 1 = √3/12*a 1 2 *h 1 .
Let the volume of the figure required by the conditions of the problem be obtained by multiplying its parameters by the coefficient k. We have:
V 2 = √3/12*k 2 *a 1 2 *k*h 1 = k 3 *V 1 .
Since the ratio of the volumes of the figures is known from the condition, we obtain the value of the coefficient k:
k = ∛(V 2 /V 1) = ∛(1/3) ≈ 0.693.
Note that we would obtain a similar value for the coefficient k for a pyramid of any type, and not just for a regular triangular one.
The main characteristic of any geometric figure in space is its volume. In this article we will look at what a pyramid with a triangle at the base is, and we will also show how to find the volume of a triangular pyramid - regular full and truncated.
What is this - a triangular pyramid?
Everyone has heard about the ancient Egyptian pyramids, but they are regular quadrangular, not triangular. Let's explain how to get a triangular pyramid.
Let's take an arbitrary triangle and connect all its vertices with some single point located outside the plane of this triangle. The resulting figure will be called a triangular pyramid. It is shown in the figure below.
As you can see, the figure in question is formed by four triangles, which in general are different. Each triangle is the sides of the pyramid or its face. This pyramid is often called a tetrahedron, that is, a tetrahedral three-dimensional figure.
In addition to the sides, the pyramid also has edges (there are 6 of them) and vertices (of 4).
with triangular base
A figure that is obtained using an arbitrary triangle and a point in space will be an irregular slanted pyramid in the general case. Now imagine that the original triangle has identical sides, and a point in space is located exactly above its geometric center at a distance h from the plane of the triangle. The pyramid constructed using these initial data will be correct.
Obviously, the number of edges, sides and vertices of a regular triangular pyramid will be the same as that of a pyramid built from an arbitrary triangle.
However, the correct figure has some distinctive features:
- its height drawn from the vertex will exactly intersect the base at the geometric center (the point of intersection of the medians);
- the lateral surface of such a pyramid is formed by three identical triangles, which are isosceles or equilateral.
A regular triangular pyramid is not only a purely theoretical geometric object. Some structures in nature have its shape, for example the diamond crystal lattice, where a carbon atom is connected to four of the same atoms by covalent bonds, or a methane molecule, where the vertices 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 the base, h is the height of the figure drawn to the marked base from the top of the pyramid.
Since the area of an arbitrary triangle is equal to half the product of the length of its side a and the apothem h a dropped onto this side, the formula for the volume of a triangular pyramid can be written in the following form:
V = 1/6 × a × h a × h
For the 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 a general equation.
For the correct one, it has a specific appearance. The area of the base (of an equilateral triangle) for it is equal to:
Substituting it into the general expression for V, we get:
V = √3/12 × a 2 × h
A special case is the situation when all sides of a tetrahedron turn out to be identical equilateral triangles. In this case, its volume can be determined only based on knowledge of the parameter of its edge a. The corresponding expression looks like:
Truncated pyramid
If the upper part containing the vertex is cut off from a regular triangular pyramid, you get a truncated figure. Unlike the original one, 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 = √3/12 × h × (A 2 + a 2 + A × a)
Here h is the height of the figure, A and a are the lengths of the sides of the large (lower) and small (upper) equilateral triangles, respectively.
The solution of the problem
To make the information in the article clearer to the reader, we will show with a clear example how to use some of the written formulas.
Let the volume of the triangular pyramid be 15 cm 3 . It is known that the figure is correct. You should find the apothem a b of the lateral edge if you know that the height of the pyramid is 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 = √3/12 × a 2 × h =>
a = 12 × V / (√3 × h) = 12 × 15 / (√3 × 4) = 25.98 cm
a b = √(h 2 + a 2 / 12) = √(16 + 25.98 2 / 12) = 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.