Volume formulas of a regular triangular pyramid. Examples of solving problems

One of the simplest three-dimensional figures is a triangular pyramid, since it consists of the smallest number of faces from which you can form a figure in space. In this article, we 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 of whose vertices are connected to one point 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 under consideration consists of a base (triangle) and three side faces (triangles). The point at which the three side faces are connected is called the top of the figure. The perpendicular from this top, lowered 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 inclined.

As was said, the base of a triangular pyramid can be a triangle of a general type. However, if it is equilateral, and the pyramid itself is straight, then they talk about the correct three-dimensional figure.

Any triangular pyramid has 4 faces, 6 edges and 4 vertices. If the lengths of all the edges are equal, then such a figure is called a tetrahedron.

Volume of a pyramid of a triangular general type

Before writing the formula for the volume of a regular triangular pyramid, we give an expression of this physical quantity for a general type pyramid. This expression has the form:

V = 1/3 * S _o * h.

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 base of the polygon of the pyramid, as well as for the cone. If at the base there is a triangle having the length of the side a and the height h _o lowered onto it, then the formula for the volume is written as follows:

V = 1/6 * a * h _o * h.

Volume formulas of a regular triangular pyramid

The 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:

h _o = √3 / 2 * a.

Substituting this expression in the formula for the volume of the 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 whose radius uniquely determines the length of the side of the polygon, then this formula can be written in terms of the corresponding radius r:

V = √3 / 4 * h * r ² .

This formula can be easily obtained from the previous one, given that the radius r of the circumscribed circle through the length of the side a of the triangle is determined by the expression:

r = a / √3.

The task of determining the volume of a tetrahedron

We show how to use the above formulas to solve specific geometry problems.

It is known that a tetrahedron has an edge length of 7 cm. Find the volume of a regular triangular tetrahedron pyramid.

Recall that the tetrahedron is a regular triangular pyramid in which all the bases are equal to each other. To use the volume formula of a regular triangular pyramid , it is necessary to calculate two quantities:

• the length of the side of the triangle;
• height of the figure.

The first quantity is known from the conditions of the problem:

a = 7 cm.

To determine the height, consider the figure shown in the figure.

The marked triangle ABC is rectangular, where the angle ABC is 90 ^o . The AC side is a hypotenuse whose length is a. Using simple geometric considerations, it can be shown that the side BC has a length:

BC = a / √3.

Note that the length BC is the radius of the circle circumscribed around the triangle.

The height of the pyramid h can be calculated using the Pythagorean theorem. We have:

h = AB = √ (AC ² - BC ² ) = √ (a ² - a 2/3) = a * √ (2/3).

Now we can substitute h and a into the corresponding formula for the volume:

V = √3 / 12 * a ² * a * √ (2/3) = √2 / 12 * a ³ .

Thus, we have obtained the tetrahedron volume formula. 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 = √2 / 12 * 7 ³ ≈ 40.42 cm ³ .

If we compare this value with the volume of the cube having the same edge, we obtain that the volume of the tetrahedron is 8.5 times smaller. This indicates that the tetrahedron is a compact figure, which is realized in some natural substances. For example, a methane molecule has a tetrahedral shape, and each carbon atom in a diamond is connected to four other atoms forming a tetrahedron.

The problem with homothetic pyramids

We will solve one curious geometric problem. Suppose that there is a triangular regular pyramid with some volume V ₁ . How many times should the size of this figure be reduced to get a pyramid homothetic to it with a volume three times smaller than the original?

We will begin to solve 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 necessary by the condition 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 condition knows the ratio of the volumes of the figures, we obtain the value of the coefficient k:

k = ∛ (V ₂ / V ₁ ) = ∛ (1/3) ≈ 0.693.

Note that we would get a similar value of the coefficient k for a pyramid of arbitrary type, and not just for a regular triangular one.