In geometry, two important characteristics are used to study figures: the lengths of the sides and the angles between them. In the case of spatial figures, dihedral angles are added to these characteristics. Let's consider what it is, and also describe the method for determining these angles using the example of a pyramid.
The concept of dihedral angle
Everyone knows that two intersecting lines form an angle with the vertex at the point of their intersection. This angle can be measured with a protractor, or you can use trigonometric functions to calculate it. The angle formed by two right angles is called linear.
Now imagine that in three-dimensional space there are two planes that intersect in a straight line. They are shown in the picture.
A dihedral angle is the angle between two intersecting planes. Just like linear, it is measured in degrees or radians. If to any point of the line along which the planes intersect, restore two perpendiculars,lying in these planes, then the angle between them will be the desired dihedral. The easiest way to determine this angle is to use the general equations of planes.
The equation of planes and the formula for the angle between them
The equation of any plane in space in general terms is written as follows:
A × x + B × y + C × z + D=0.
Here x, y, z are the coordinates of points belonging to the plane, the coefficients A, B, C, D are some known numbers. The convenience of this equality for calculating dihedral angles is that it explicitly contains the coordinates of the direction vector of the plane. We will denote it by n¯. Then:
n¯=(A; B; C).
The vector n¯ is perpendicular to the plane. The angle between two planes is equal to the angle between their direction vectors n1¯ and n2¯. It is known from mathematics that the angle formed by two vectors is uniquely determined from their scalar product. This allows you to write a formula for calculating the dihedral angle between two planes:
φ=arccos (|(n1¯ × n2¯)| / (|n1 ¯| × |n2¯|)).
If we substitute the coordinates of the vectors, the formula will be written explicitly:
φ=arccos (|A1 × A2 + B1 × B 2 + C1 × C2| / (√(A1 2 + B12 + C12 ) × √(A22+B22 + C22))).
The modulo sign in the numerator is used to define only an acute angle, since a dihedral angle is always less than or equal to 90o.
Pyramid and its corners
Pyramid is a figure formed by one n-gon and n triangles. Here n is an integer equal to the number of sides of the polygon that is the base of the pyramid. This spatial figure is a polyhedron or polyhedron, since it consists of flat faces (sides).
The dihedral angles of a pyramid-polyhedron can be of two types:
- between base and side (triangle);
- between two sides.
If the pyramid is considered regular, then it is easy to determine the named angles for it. To do this, using the coordinates of three known points, one should compose an equation of planes, and then use the formula given in the paragraph above for the angle φ.
Below we give an example in which we show how to find dihedral angles at the base of a quadrangular regular pyramid.
A quadrangular regular pyramid and an angle at its base
Assume that a regular pyramid with a square base is given. The length of the side of the square is a, the height of the figure is h. Find the angle between the base of the pyramid and its side.
Let's place the origin of the coordinate system in the center of the square. Then the coordinates of the pointsA, B, C, D shown in the picture will be:
A=(a/2; -a/2; 0);
B=(a/2; a/2; 0);
C=(-a/2; a/2; 0);
D=(0; 0; h).
Consider the planes ACB and ADB. Obviously, the direction vector n1¯ for the ACB plane will be:
1¯=(0; 0; 1).
To determine the direction vector n2¯ of the ADB plane, proceed as follows: find two arbitrary vectors that belong to it, for example, AD¯ and AB¯, then calculate their vector work. Its result will give the coordinates n2¯. We have:
AD¯=D - A=(0; 0; h) - (a/2; -a/2; 0)=(-a/2; a/2; h);
AB¯=B - A=(a/2; a/2; 0) - (a/2; -a/2; 0)=(0; a; 0);
2¯=[AD¯ × AB¯]=[(-a/2; a/2; h) × (0; a; 0)]=(-a × h; 0;-a2/2).
Since multiplication and division of a vector by a number does not change its direction, we transform the resulting n2¯, dividing its coordinates by -a, we get:
2¯=(h; 0; a/2).
We have defined vector guides n1¯ and n2¯ for the ACB base and ADB side planes. It remains to use the formula for the angle φ:
φ=arccos (|(n1¯ × n2¯)| / (|n1 ¯| × |n2¯|))=arccos (a / (2 × √h2 + a 2/4)).
Transform the resulting expression and rewrite it like this:
φ=arccos (a / √(a2+ 4 × h2)).
We have obtained the formula for the dihedral angle at the base for a regular quadrangular pyramid. Knowing the height of the figure and the length of its side, you can calculate the angle φ. For example, for the pyramid of Cheops, whose base side is 230.4 meters, and the initial height was 146.5 meters, the angle φ will be 51.8o.
It is also possible to determine the dihedral angle for a quadrangular regular pyramid using the geometric method. To do this, it suffices to consider a right-angled triangle formed by height h, half the length of the base a/2 and the apothem of an isosceles triangle.
