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. Consider what it is, and also describe the methodology for determining these angles using the example of a pyramid.
The concept of dihedral angle
Everyone knows that two intersecting lines form a certain angle with a vertex at the point of intersection. This angle can be measured using a protractor or use the trigonometric functions to calculate it. The angle formed by two straight lines is called linear.
Now imagine that in three-dimensional space there are two planes that intersect in a straight line. They are depicted in the figure.
The dihedral angle is the angle between two intersecting planes. Like linear, it is measured in degrees or radians. If two perpendiculars lying in these planes are restored to any point on the line along which the planes intersect, then the angle between them will be the desired dihedral. The easiest way to determine this angle is to use the equations of the planes in general form.
The equation of planes and the formula for the angle between them
The equation of any plane in space is generally 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 well-known numbers. The convenience of this equality for calculating dihedral angles lies in the fact that it explicitly contains the coordinates of the directing vector of the plane. We denote it by n¯. Then:
n¯ = (A; B; C).
Vector n¯ is perpendicular to the plane. The angle between two planes is equal to the angle between their guide vectors n 1 ¯ and n 2 ¯. 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 (| (n 1 ¯ × n 2 ¯) | / (| n 1 ¯ | × | n 2 ¯ |)).
If we substitute the coordinates of the vectors, then the formula is written in explicit form:
φ = arccos (| A 1 × A 2 + B 1 × B 2 + C 1 × C 2 | / (√ (A 1 2 + B 1 2 + C 1 2 ) × √ (A 2 2 + B 2 2 + C 2 2 ))).
The sign of the module in the numerator is used to determine only an acute angle, since the dihedral angle is always less than or equal to 90 o .
Pyramid and its corners
A pyramid is a figure that is formed by one n-gon and n triangles. Here n is an integer equal to the number of sides of the polygon, which is the base of the pyramid. This spatial figure is a polyhedron or polyhedron, because it consists of flat faces (sides).
The dihedral angles of a polyhedron-pyramid can be of two types:
- between the base and the side (triangle);
- between two sides.
If the pyramid is considered correct, then the named angles for it are easy to determine. To do this, according to the coordinates of three known points, one should draw up an equation of planes, and then use the formula given above for the angle φ.
Below is an example in which we show how to find dihedral angles at the base of a regular pyramid.
The quadrangular regular pyramid and the angle at its base
Suppose you are given a regular pyramid with a square base. 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.
We place the beginning of the coordinate system in the center of the square. Then the coordinates of points A, B, C, D shown in the figure will be equal to:
A = (a / 2; -a / 2; 0);
B = (a / 2; a / 2; 0);
C = (-a / 2; a / 2; 0);
D = (0; 0; h).
Consider the ACB and ADB planes. Obviously, the direction vector n 1 ¯ for the ACB plane will be equal to:
n 1 ¯ = (0; 0; 1).
To determine the direction vector n 2 ¯ of the ADB plane, we proceed as follows: we find two arbitrary vectors that belong to it, for example, AD¯ and AB¯, then we calculate their vector product. Its result will give the coordinates n 2 ¯. 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);
n 2 ¯ = [AD¯ × AB¯] = [(-a / 2; a / 2; h) × (0; a; 0)] = (-a × h; 0; -a 2/2).
Since multiplying and dividing a vector by a number does not change its direction, we transform the resulting n 2 ¯, dividing its coordinates by -a, we get:
n 2 ¯ = (h; 0; a / 2).
We determined the guiding vectors n 1 ¯ and n 2 ¯ for the base planes of ACB and the lateral side of ADB. It remains to use the formula for the angle φ:
φ = arccos (| (n 1 ¯ × n 2 ¯) | / (| n 1 ¯ | × | n 2 ¯ |)) = arccos (a / (2 × √h 2 + a 2/4)).
We transform the resulting expression and rewrite it like this:
φ = arccos (a / √ (a 2 + 4 × h 2 )).
We got the formula for the dihedral angle at the base for the regular quadrangular pyramid. Knowing the height of the figure and the length of its side, we can calculate the angle φ. For example, for the Cheops pyramid, the base side of which is 230.4 meters, and the initial height was 146.5 meters, the angle φ will be 51.8 o .
The dihedral angle for a quadrangular regular pyramid can also be determined using the geometric method. To do this, it is enough to consider a right-angled triangle formed by a height h, half the base length a / 2 and the apothem of an isosceles triangle.