How to calculate the volume of a pyramid by the coordinates of the vertices? Methodology and example problem

Often in the problems of the school course of geometry, one has to solve tasks that require the use of an integrated approach. One such task is to calculate the volume of a pyramid based on the coordinates of the vertices. How to solve this geometric problem - the following article will answer.

What is a pyramid?

In simple words, this figure refers to a spatial object bounded by triangular sides and one polygonal face, which is called the base. The polygonal base can be an arbitrary n-gon in the plane, for example, a regular triangle, parallelogram, and so on.

Any pyramid has n + 1 faces, 2 * n edges and n + 1 vertices. The vertices of the figure are not equal. So, there is a single peak that does not belong to the base. It is called the main one. The distance from it to the base plane is the height of the figure.

Pyramids can be inclined if the height does not cross the base at its center, or straight when the height intersects the base at the geometric center of the latter. Also shapes can be wrong and right. Regular pyramids consist of an equilateral and equilateral base and several isosceles triangles that are equal to each other.

How is the volume of the pyramid calculated?

Before we give a methodology for calculating the coordinates of the vertices of the volume of the pyramid, you should give a formula with which you can calculate this value for a figure of any type from the class in question. So, the volume of the pyramid is calculated as follows:

V = 1/3 * S _o * h.

Here S _o is the base area, h is the distance from the main peak to the base, that is, the height of the pyramid.

Thus, any geometric problem of finding the volume of a pyramid is reduced to calculating the values ​​of S _o and h.

How to find the volume of a pyramid by the coordinates of the vertices: a technique

A pyramid can be represented by an arbitrary n-coal base. In order to calculate its area, one should carefully study the condition of the problem, in which it should be said what type of n-gon is involved. If it is a triangle or parallelogram, then calculating its area by known coordinates is very simple: you just need to find the vector product of the corresponding side vectors.

Calculating the height of the pyramid is also not difficult. To do this, it follows from any three points of the base to obtain the plane equation in general form, and then you need to use the distance formula between the plane and the point (the top of the pyramid). The formula is:

d = | (A * x ₁ + B * y ₁ + C * z ₁ + D) | / √ (A ² + B ² + C ² ).

Here (x ₁ ; y ₁ ; z ₁ ) are the coordinates of the point.

The equation of the plane has the form:

A * x + B * y + C * z + D = 0.

The task with a triangular pyramid

Let's solve the problem using the simplest pyramid as a triangular one. The condition is simple: the coordinates of the vertices of the pyramid are given below, the volume must be found for the figure, which is built on these coordinates:

• A (1; 0; 3);
• B (0; 2; -1);
• C (3; 3; 1);
• D (4; 3; 4).

Suppose that the base of the pyramid is a triangle ABC. Find the lengths of the vectors AB¯ and AC¯:

AB¯ = (-1; 2; -4);

AC¯ = (2; 3; -2).

The vector product AB¯ and AC¯ gives us, on the one hand, the double area of ​​the triangle, that is 2 * S _o , and on the other hand, we get the coordinates of the vector n¯ normal to the plane, we have:

n¯ = [AB¯ * AC¯] = (8; -10; -7).

The area of ​​the triangular base is equal to the half-length of the vector n¯, that is:

S _o = √ (8 ² + 10 ² + 7 ² ) / 2 = 7.3.

Before calculating the distance from D to the plane ABC, it is necessary to write the equation of the plane. Three of its coefficients (A, B, C) we already know, they correspond to the coordinates of the normal n¯. A free term can be obtained by substituting in the equation the coordinates of any point on the plane, for example, point A, we have:

D = -1 * (A * x ₁ + B * y ₁ + C * z ₁ ) = -1 * (8 * 1 + (-10) * 0 + (-7) * 3) = 13.

Then the equation of the plane of the base of the pyramid takes the form:

8 * x - 10 * y - 7 * z + 13 = 0.

Now we apply the above formula to calculate the distance from the point D (4; 3; 4) to the found plane, we get:

d = | (8 * 4 - 10 * 3 - 7 * 4 + 13) | / √ (8 ² + 10 ² + 7 ² ) = 0.89.

Since the found value of the distance d corresponds to the height of the triangular pyramid h, then we can use the formula for the volume of the figure:

V = 1/3 * S _o * h = 1/3 * 7.3 * 0.89 ≈ 2.166.

The obtained volume value is expressed in cubic units of the selected coordinate system.