How to find the cosine of an angle between planes. Dihedral angle

This article is about the angle between planes and how to find it. First, the definition of the angle between two planes is given and a graphic illustration is given. After that, the principle of finding the angle between two intersecting planes by the coordinate method was analyzed, a formula was obtained that allows calculating the angle between intersecting planes using the known coordinates of the normal vectors of these planes. In conclusion, detailed solutions of typical problems are shown.

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Angle between planes - definition.

Let us give arguments that will allow us to gradually approach the definition of the angle between two intersecting planes.

Let us be given two intersecting planes and . These planes intersect in a straight line, which we denote by the letter c. Let's construct a plane passing through the point M of the line c and perpendicular to the line c. In this case, the plane will intersect the planes and . Denote the line along which the planes intersect and as a, and the line along which the planes intersect and as b. Obviously, the lines a and b intersect at the point M.

It is easy to show that the angle between the intersecting lines a and b does not depend on the location of the point M on the line c through which the plane passes.

Let us construct a plane perpendicular to the line c and different from the plane . The plane is intersected by the planes and along straight lines, which we denote by a 1 and b 1, respectively.

From the method of constructing planes and it follows that the lines a and b are perpendicular to the line c, and the lines a 1 and b 1 are perpendicular to the line c. Since the lines a and a 1 lie in the same plane and are perpendicular to the line c, they are parallel. Similarly, lines b and b 1 lie in the same plane and are perpendicular to line c, hence they are parallel. Thus, it is possible to perform a parallel transfer of the plane to the plane, in which the line a 1 coincides with the line a, and the line b with the line b 1. Therefore, the angle between two intersecting lines a 1 and b 1 is equal to the angle between intersecting lines a and b .

This proves that the angle between the intersecting lines a and b lying in the intersecting planes and does not depend on the choice of the point M through which the plane passes. Therefore, it is logical to take this angle as the angle between two intersecting planes.

Now you can voice the definition of the angle between two intersecting planes and .

Definition.

The angle between two planes intersecting in a straight line and is the angle between two intersecting lines a and b, along which the planes and intersect with the plane perpendicular to the line c.

The definition of the angle between two planes can be given a little differently. If on the line c, along which the planes intersect, mark the point M and draw lines through it a and b, perpendicular to the line c and lying in the planes and, respectively, then the angle between the lines a and b is the angle between the planes and. Usually, in practice, such constructions are performed in order to obtain the angle between the planes.

Since the angle between the intersecting lines does not exceed , it follows from the voiced definition that the degree measure of the angle between two intersecting planes is expressed by a real number from the interval . In this case, intersecting planes are called perpendicular if the angle between them is ninety degrees. The angle between parallel planes is either not determined at all, or it is considered equal to zero.

Finding the angle between two intersecting planes.

Usually, when finding the angle between two intersecting planes, you first have to perform additional constructions in order to see the intersecting lines, the angle between which is equal to the desired angle, and then connect this angle with the original data using equal signs, similarity signs, the cosine theorem or the definitions of sine, cosine and the tangent of the angle. In the geometry course of high school, there are similar problems.

For example, let's give a solution to problem C2 from the Unified State Examination in mathematics for 2012 (the condition is intentionally changed, but this does not affect the principle of the solution). In it, it was just necessary to find the angle between two intersecting planes.

Example.

Decision.

First, let's make a drawing.

Let's perform additional constructions to "see" the angle between the planes.

First, let's define a straight line along which the planes ABC and BED 1 intersect. Point B is one of their common points. Find the second common point of these planes. The lines DA and D 1 E lie in the same plane ADD 1, and they are not parallel, and, therefore, intersect. On the other hand, the line DA lies in the plane ABC, and the line D 1 E lies in the plane BED 1, therefore, the intersection point of the lines DA and D 1 E will be a common point of the planes ABC and BED 1. So, we continue the lines DA and D 1 E until they intersect, we denote the point of their intersection with the letter F. Then BF is the straight line along which the planes ABC and BED 1 intersect.

It remains to construct two lines lying in the planes ABC and BED 1, respectively, passing through one point on the line BF and perpendicular to the line BF - the angle between these lines, by definition, will be equal to the desired angle between the planes ABC and BED 1 . Let's do it.

Dot A is the projection of the point E onto the plane ABC. Draw a line that intersects at a right angle the line BF at the point M. Then the line AM is the projection of the line EM onto the plane ABC, and by the three perpendiculars theorem.

Thus, the desired angle between the planes ABC and BED 1 is .

We can determine the sine, cosine or tangent of this angle (and hence the angle itself) from a right triangle AEMif we know the lengths of its two sides. From the condition it is easy to find the length AE: since point E divides side AA 1 in relation to 4 to 3, counting from point A, and the length of side AA 1 is 7, then AE \u003d 4. Let's find the length of AM.

To do this, consider a right triangle ABF with right angle A, where AM is the height. By condition AB=2. We can find the length of the side AF from the similarity of right triangles DD 1 F and AEF :

By the Pythagorean theorem, from the triangle ABF we find . We find the length AM through the area of the triangle ABF: on one side, the area of the triangle ABF is equal to , on the other side , where .

Thus, from the right triangle AEM we have .

Then the desired angle between the planes ABC and BED 1 is (note that ).

Answer:

In some cases, to find the angle between two intersecting planes, it is convenient to specify Oxyz and use the coordinate method. Let's stop on it.

Let's set the task: to find the angle between two intersecting planes and . Let's denote the desired angle as .

We will assume that in a given rectangular coordinate system Oxyz we know the coordinates of the normal vectors of the intersecting planes and or it is possible to find them. Let be is the normal vector of the plane, and is the normal vector of the plane . Let us show how to find the angle between intersecting planes and through the coordinates of the normal vectors of these planes.

Let us denote the line along which the planes intersect and as c . Through the point M on the line c we draw a plane perpendicular to the line c. The plane intersects the planes and along the lines a and b, respectively, the lines a and b intersect at the point M. By definition, the angle between intersecting planes and is equal to the angle between intersecting lines a and b.

Let us set aside from the point M in the plane the normal vectors and of the planes and . In this case, the vector lies on a line that is perpendicular to line a, and the vector lies on a line that is perpendicular to line b. Thus, in the plane, the vector is the normal vector of the line a, is the normal vector of the line b.

In the article Finding the angle between intersecting lines, we obtained a formula that allows you to calculate the cosine of the angle between intersecting lines using the coordinates of normal vectors. Thus, the cosine of the angle between the lines a and b, and, consequently, and cosine of the angle between intersecting planes and is found by the formula , where and are the normal vectors of the planes and, respectively. Then it is calculated as .

Let's solve the previous example using the coordinate method.

Example.

A rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 is given, in which AB \u003d 2, AD \u003d 3, AA 1 \u003d 7 and point E divides side AA 1 in a ratio of 4 to 3, counting from point A. Find the angle between the planes ABC and BED 1.

Decision.

Since the sides of a rectangular parallelepiped at one vertex are pairwise perpendicular, it is convenient to introduce a rectangular coordinate system Oxyz as follows: the beginning is aligned with the vertex C, and the coordinate axes Ox, Oy and Oz are directed along the sides CD, CB and CC 1, respectively.

The angle between the planes ABC and BED 1 can be found through the coordinates of the normal vectors of these planes using the formula , where and are the normal vectors of the planes ABC and BED 1, respectively. Let us determine the coordinates of normal vectors.

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Theorem

The angle between planes does not depend on the choice of cutting plane.

Proof.

Let there be two planes α and β that intersect along the line c. draw the plane γ perpendicular to the line c. Then the plane γ intersects the planes α and β along the lines a and b, respectively. The angle between the planes α and β is equal to the angle between the lines a and b.
Take another cutting plane γ`, perpendicular to c. Then the plane γ` will intersect the planes α and β along the lines a` and b` respectively.
With parallel translation, the point of intersection of the plane γ with the line c will go to the point of intersection of the plane γ` with the line c. in this case, by the property of parallel translation, the line a will go to the line a`, b - to the line b`. hence the angles between lines a and b, a` and b` are equal. The theorem has been proven.

Page navigation.

Angle between planes - definition.

When presenting the material, we will use the definitions and concepts given in the articles plane in space and straight line in space.

Let us give arguments that will allow us to gradually approach the definition of the angle between two intersecting planes.

Let us be given two intersecting planes and . These planes intersect in a straight line, which we denote by the letter c. Construct a plane passing through the point M straight c and perpendicular to the line c. In this case, the plane will intersect the planes and . We denote the line along which the planes intersect and as a, but the straight line along which the planes intersect and how b. Obviously direct. a and b intersect at a point M.

It is easy to show that the angle between intersecting lines a and b does not depend on the location of the point M on a straight line c through which the plane passes.

Construct a plane perpendicular to the line c and different from the plane. The plane is intersected by planes and along straight lines, which we denote a 1 and b 1 respectively.

It follows from the method of constructing planes that the lines a and b perpendicular to the line c, and direct a 1 and b 1 perpendicular to the line c. Since straight a and a 1 c, then they are parallel. Likewise, straight b and b 1 lie in the same plane and are perpendicular to the line c so they are parallel. Thus, it is possible to perform a parallel transfer of the plane to the plane, in which the straight line a 1 coincides with the line a, and the straight line b with a straight line b 1. Therefore, the angle between two intersecting lines a 1 and b 1 equal to the angle between intersecting lines a and b.

This proves that the angle between intersecting lines a and b lying in intersecting planes and does not depend on the choice of the point M through which the plane passes. Therefore, it is logical to take this angle as the angle between two intersecting planes.

Now you can voice the definition of the angle between two intersecting planes and .

Definition.

Angle between two intersecting lines c planes and is the angle between two intersecting lines a and b, along which the planes and intersect with the plane perpendicular to the line c.

The definition of the angle between two planes can be given a little differently. If on a straight line with, along which the planes and intersect, mark a point M and draw straight lines through it a and b, perpendicular to the line c and lying in the planes and respectively, then the angle between the lines a and b is the angle between the planes and . Usually, in practice, such constructions are performed in order to obtain the angle between the planes.

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Finding the angle between two intersecting planes.

ABCDA 1 B 1 C 1 D 1, wherein AB=3, AD=2, AA 1 =7 and dot E divides the side AA 1 in a relationship 4 to 3 , counting from the point BUT ABC and BED 1.

First, let's make a drawing.

Let's perform additional constructions to "see" the angle between the planes.

First, we define a straight line along which the planes intersect ABC and Bed 1. Dot AT is one of their common points. Find the second common point of these planes. Direct DA and D 1 E lie in the same plane ADD 1, and they are not parallel, and, therefore, intersect. On the other hand, straight DA lies in the plane ABC, and the straight line D 1 E- in the plane Bed 1, hence the point of intersection of the lines DA and D 1 E will be a common point of the planes ABC and Bed 1. So let's continue straight DA and D 1 E before they intersect, we denote the point of their intersection by the letter F. Then bf- a line along which the planes intersect ABC and Bed 1.

It remains to construct two straight lines lying in planes ABC and Bed 1 respectively, passing through one point on the line bf and perpendicular to the line bf, - the angle between these lines, by definition, will be equal to the desired angle between the planes ABC and Bed 1. Let's do it.

Dot BUT is the projection of the point E to the plane ABC. Draw a line that intersects the line at right angles BF at the point M. Then the line AM is a projection of a straight line EAT to the plane ABC, and by the three perpendiculars theorem.

Thus, the desired angle between the planes ABC and Bed 1 is equal to .

The sine, cosine or tangent of this angle (and hence the angle itself) we can determine from a right triangle AEM if we know the lengths of its two sides. From the condition it is easy to find the length AE: since dot E divides the side AA 1 in a relationship 4 to 3 , counting from the point BUT, and the side length AA 1 is equal to 7 , then AE=4. Let's find another length AM.

To do this, consider a right triangle ABF right angle BUT, where AM is the height. By condition AB=2. side length AF we can find from the similarity of right triangles DD 1F and AEF:

By the Pythagorean theorem from a triangle ABF find . Length AM find through the area of a triangle ABF: on one side the area of a triangle ABF is equal to , on the other hand , whence .

So from a right triangle AEM we have .

Then the desired angle between the planes ABC and Bed 1 equals (note that ).

In some cases, to find the angle between two intersecting planes, it is convenient to set a rectangular coordinate system Oxyz and use the coordinate method. Let's stop on it.

Let's set the task: to find the angle between two intersecting planes and . Let's denote the desired angle as .

We assume that in a given rectangular coordinate system Oxyz we know the coordinates of the normal vectors of the intersecting planes and or have the opportunity to find them. Let be a normal vector of the plane , and be a normal vector of the plane . Let us show how to find the angle between intersecting planes and through the coordinates of the normal vectors of these planes.

Let us denote the line along which the planes intersect and as c. Through the dot M on a straight line c draw a plane perpendicular to the line c. The plane intersects planes and along straight lines a and b respectively, direct a and b intersect at a point M. By definition, the angle between intersecting planes and is equal to the angle between intersecting lines a and b.

Set aside from the point M in the plane are the normal vectors and of the planes and . The vector lies on a line that is perpendicular to the line a, and the vector is on a line that is perpendicular to the line b. Thus, in the plane, the vector is the normal vector of the line a, - normal line vector b.

In the article Finding the angle between intersecting lines, we obtained a formula that allows you to calculate the cosine of the angle between intersecting lines using the coordinates of normal vectors. So the cosine of the angle between the lines a and b, and, consequently, cosine of the angle between intersecting planes and is found by the formula , where and are the normal vectors of the planes and, respectively. Then angle between intersecting planes is calculated as .

Let's solve the previous example using the coordinate method.

Given a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1, wherein AB=3, AD=2, AA 1 =7 and dot E divides the side AA 1 in a relationship 4 to 3 , counting from the point BUT. Find the angle between planes ABC and BED 1.

Since the sides of a rectangular parallelepiped at one vertex are pairwise perpendicular, it is convenient to introduce a rectangular coordinate system Oxyz like this: start to combine with the top With, and the coordinate axes Ox, Oy and Oz send around CD, CB and CC 1 respectively.

Angle between planes ABC and Bed 1 can be found through the coordinates of the normal vectors of these planes by the formula , where and are the normal vectors of the planes ABC and Bed 1 respectively. Let us determine the coordinates of normal vectors.

Since the plane ABC coincides with the coordinate plane Oxy, then its normal vector is the coordinate vector , that is, .

As a normal plane vector Bed 1 we can take the cross product of vectors and , in turn, the coordinates of the vectors and can be found through the coordinates of the points AT, E and D1(which is written in the article the coordinates of the vector through the coordinates of the points of its beginning and end), and the coordinates of the points AT, E and D1 in the introduced coordinate system, we determine from the condition of the problem.

Obviously, . Since , then we find by the coordinates of the points (if necessary, see the article division of a segment in a given ratio). Then and Oxyz are equations and .

When we studied the general equation of a straight line, we found out that the coefficients BUT, AT and With are the corresponding coordinates of the normal vector of the plane. Thus, and are the normal vectors of the planes and, respectively.

We substitute the coordinates of the normal vectors of the planes into the formula for calculating the angle between two intersecting planes:

Then . Since the angle between two intersecting planes is not obtuse, then using the basic trigonometric identity we find the sine of the angle:.

The measure of the angle between planes is the acute angle formed by two straight lines lying in these planes and drawn perpendicular to the line of their intersection.

Construction algorithm

From an arbitrary point K, perpendiculars are drawn to each of the given planes.
The rotation around the level line determines the value of the angle γ° with the apex at the point K.
Calculate the angle between the planes ϕ° = 180 - γ° provided that γ° > 90°. If γ°< 90°, то ∠ϕ° = ∠γ°.

The figure shows the case when the planes α and β are given by traces. All necessary constructions are made according to the algorithm and are described below.

Decision

In an arbitrary place of the drawing, we mark the point K. From it we lower the perpendiculars m and n, respectively, to the planes α and β. The direction of the projections m and n is as follows: m""⊥f 0α , m"⊥h 0α , n""⊥f 0β , n"⊥h 0β .
We determine the actual size ∠γ° between the lines m and n. To do this, rotate the angle plane with vertex K around the frontal f to a position parallel to the frontal projection plane. The turning radius R of the point K is equal to the value of the hypotenuse of the right triangle O""K""K 0 , whose leg is K""K 0 = y K – y O .
The desired angle is ϕ° = ∠γ°, since ∠γ° is acute.

The figure below shows the solution to the problem in which it is required to find the angle γ° between the planes α and β, given by parallel and intersecting lines, respectively.

Decision

We determine the direction of the projections of the horizontals h 1 , h 2 and the frontals f 1 , f 2 belonging to the planes α and β, in the order indicated by the arrows. From an arbitrary point K on the square. α and β we drop the perpendiculars e and k. In this case, e""⊥f"" 1 , e"⊥h" 1 and k""⊥f"" 2 , k"⊥h" 2 .
We determine ∠γ° between lines e and k. To do this, we draw a horizontal h 3 and rotate the point K around it to the position K 1, at which △CKD will become parallel to the horizontal plane and be reflected on it in full size - △C "K" 1 D". The projection of the center of rotation O" is on the drawn to h "3 perpendicular K "O". Radius R is determined from a right triangle O "K" K 0, whose side is K "K 0 \u003d Z O - Z K.
The desired value is ∠ϕ° = ∠γ°, since the angle γ° is acute.

When solving geometric problems in space, there are often those where it is necessary to calculate the angles between different spatial objects. In this article, we will consider the question of finding the angles between planes and between them and a straight line.

Straight line in space

It is known that absolutely any straight line in the plane can be defined by the following equality:

Here a and b are some numbers. If we represent a straight line in space with the same expression, then we get a plane parallel to the z axis. For the mathematical definition of the spatial line, a different solution method is used than in the two-dimensional case. It consists in using the concept of "directing vector".

Examples of solving problems for determining the angle of intersection of planes

Knowing how to find the angle between the planes, we will solve the following problem. Two planes are given, the equations of which have the form:

3 * x + 4 * y - z + 3 = 0;
X - 2 * y + 5 * z +1 = 0

What is the angle between the planes?

To answer the question of the problem, we recall that the coefficients that stand at the variables in the general equation of the plane are the coordinates of the guide vector. For these planes, we have the following coordinates of their normals:

n 1 ¯(3; 4; -1);
n 2 ¯(-1; -2; 5)

Now we find the scalar product of these vectors and their modules, we have:

(n 1 ¯ * n 2 ¯) \u003d -3 -8 -5 \u003d -16;
|n 1 ¯| = √(9 + 16 + 1) = √26;
|n 2 ¯| = √(1 + 4 + 25) = √30

Now you can substitute the found numbers into the formula given in the previous paragraph. We get:

α = arccos(|-16 | / (√26 * √30) ≈ 55.05 o

The resulting value corresponds to the acute angle of intersection of the planes specified in the condition of the problem.

Now let's look at another example. Given two planes:

Do they intersect? Let's write out the values of the coordinates of their direction vectors, calculate their scalar product and modules:

n 1 ¯(1; 1; 0);
n 2 ¯(3; 3; 0);
(n 1 ¯ * n 2 ¯) = 3 + 3 + 0 = 6;
|n 1 ¯| = √2;
|n 2 ¯| = √18

Then the angle of intersection is:

α = arccos(|6| / (√2 * √18) =0 o .

This angle indicates that the planes do not intersect, but are parallel. The fact that they do not match each other is easy to check. Let's take for this an arbitrary point belonging to the first of them, for example, P(0; 3; 2). Substituting its coordinates into the second equation, we get:

3 * 0 +3 * 3 + 8 = 17 ≠ 0

That is, the point P belongs only to the first plane.

Thus, two planes are parallel when their normals are.

Plane and line

In the case of considering the relative position between a plane and a straight line, there are several more options than with two planes. This fact is connected with the fact that the straight line is a one-dimensional object. Line and plane can be:

mutually parallel, in this case the plane does not intersect the line;
the latter may belong to the plane, while it will also be parallel to it;
both objects can intersect at some angle.

Consider first the last case, since it requires the introduction of the concept of the angle of intersection.

Line and plane, the value of the angle between them

If a straight line intersects a plane, then it is called inclined with respect to it. The point of intersection is called the base of the slope. To determine the angle between these geometric objects, it is necessary to lower a straight perpendicular to the plane from any point. Then the point of intersection of the perpendicular with the plane and the place of intersection of the oblique with it form a straight line. The latter is called the projection of the original line onto the plane under consideration. Acute and its projection is the desired one.

The somewhat confusing definition of the angle between a plane and an oblique will be clarified by the figure below.

Here the angle ABO is the angle between the line AB and the plane a.

To write a formula for it, consider an example. Let there be a straight line and a plane, which are described by the equations:

(x ; y ; z) = (x 0 ; y 0 ; z 0) + λ * (a; b; c);
A * x + B * x + C * x + D = 0

It is easy to calculate the desired angle for these objects if you find the scalar product between the direction vectors of the line and the plane. The resulting acute angle should be subtracted from 90 o, then it is obtained between a straight line and a plane.

The figure above demonstrates the described algorithm for finding the considered angle. Here β is the angle between the normal and the line, and α is between the line and its projection onto the plane. It can be seen that their sum is equal to 90 o .

Above, a formula was presented that answers the question of how to find an angle between planes. Now we give the corresponding expression for the case of a straight line and a plane:

α = arcsin(|a * A + b * B + c * C| / (√(a 2 + b 2 + c 2) * √(A 2 + B 2 + C 2)))

The modulus in the formula allows only acute angles to be calculated. The arcsine function appeared instead of the arccosine due to the use of the corresponding reduction formula between trigonometric functions (cos(β) = sin(90 o-β) = sin(α)).

Problem: A plane intersects a line

Now we will show how to work with the above formula. Let's solve the problem: it is necessary to calculate the angle between the y-axis and the plane given by the equation:

This plane is shown in the figure.

It can be seen that it intersects the y and z axes at the points (0; -12; 0) and (0; 0; 12), respectively, and is parallel to the x axis.

The direction vector of the straight line y has coordinates (0; 1; 0). A vector perpendicular to a given plane is characterized by coordinates (0; 1; -1). We apply the formula for the angle of intersection of a straight line and a plane, we get:

α = arcsin(|1| / (√1 * √2)) = arcsin(1 / √2) = 45o

Problem: straight line parallel to the plane

Now let's solve a problem similar to the previous one, the question of which is posed differently. The equations of the plane and the straight line are known:

x + y - z - 3 = 0;
(x; y; z) = (1; 0; 0) + λ * (0; 2; 2)

It is necessary to find out whether these geometric objects are parallel to each other.

We have two vectors: the directing line is (0; 2; 2) and the directing plane is (1; 1; -1). We find their scalar product:

0 * 1 + 1 * 2 - 1 * 2 = 0

The resulting zero indicates that the angle between these vectors is 90 o , which proves the parallelism of the straight line and the plane.

Now let's check whether this line is only parallel or also lies in a plane. To do this, select an arbitrary point on the line and check whether it belongs to the plane. For example, let's take λ = 0, then the point P(1; 0; 0) belongs to the line. We substitute into the equation of the plane P:

The point P does not belong to the plane, and hence the whole line does not lie in it.

Where is it important to know the angles between the considered geometric objects?

The above formulas and examples of problem solving are not only of theoretical interest. They are often used to determine important physical quantities of real three-dimensional figures, such as prisms or pyramids. It is important to be able to determine the angle between the planes when calculating the volumes of figures and the areas of their surfaces. Moreover, if in the case of a straight prism it is possible not to use these formulas to determine the indicated quantities, then for any type of pyramid their use is inevitable.

Below we will consider an example of using the stated theory to determine the angles of a pyramid with a square base.

Pyramid and its corners

The figure below shows a pyramid, at the base of which lies a square with side a. The height of the figure is h. You need to find two corners:

between the side surface and the base;
between the lateral edge and the base.

To solve the problem, you must first enter the coordinate system and determine the parameters of the corresponding vertices. The figure shows that the origin of coordinates coincides with the point in the center of the square base. In this case, the base plane is described by the equation:

That is, for any x and y, the value of the third coordinate is always zero. The lateral plane ABC intersects the z-axis at the point B(0; 0; h), and the y-axis at the point with coordinates (0; a/2; 0). It does not cross the x-axis. This means that the equation of the ABC plane can be written as:

y / (a / 2) + z / h = 1 or
2 * h * y + a * z - a * h = 0

The vector AB¯ is a side edge. Its start and end coordinates are: A(a/2; a/2; 0) and B(0; 0; h). Then the coordinates of the vector itself:

We have found all the necessary equations and vectors. Now it remains to use the considered formulas.

First, in the pyramid, we calculate the angle between the planes of the base and the side. The corresponding normal vectors are: n 1 ¯(0; 0; 1) and n 2 ¯(0; 2*h; a). Then the angle will be: