Typical linear parameters of any pyramid are the lengths of the sides of its base, height, side ribs and apofemas. Nevertheless, there is another characteristic that is associated with the marked parameters - it is a dihedral angle. Consider in the article what it is and how to find it.
Spatial figure pyramid
Each student is well aware of what is being said when he hears the word "pyramid". Geometrically, it can be constructed as follows: select a certain polygon, then fix a point in space and connect it with each corner of the polygon. The resulting three-dimensional figure will be a pyramid of any type. The polygon that forms it is called the base, and the point with which all its angles are connected is the top of the figure. The figure below schematically shows a pentagonal pyramid.
It can be seen that its surface is formed not only by the pentagon, but also by five triangles. In the general case, the number of these triangles will be equal to the number of sides of the polygonal base.
Dihedral corners of a figure
When considering geometric problems on the plane, then any angle is formed by two intersecting straight lines, or segments. In space, dihedral, formed by the intersection of two planes, are added to these linear angles.
If the marked definition of an angle in space is applied to the figure in question, then we can say that there are two types of dihedral angles:
- At the base of the pyramid. It is formed by the plane of the base and any of the side faces (triangle). This means that the angles at the base of the pyramid are n, where n is the number of sides of the polygon.
- Between the sides (triangles). The number of these dihedral angles is also n pieces.
Note that the first type of angles under consideration is built on the edges of the base, the second type - on the side edges.
How to calculate the corners of a pyramid?
The linear angle of the dihedral angle is a measure of the latter. It is not easy to calculate it, since the faces of the pyramid, unlike the faces of the prism, do not intersect at right angles in the general case. It is most reliable to calculate the values of dihedral angles using the equations of the plane in a general form.
In three-dimensional space, a plane is defined by the following expression:
A * x + B * y + C * z + D = 0
Where A, B, C, D are some real numbers. The convenience of this equation is that the first three marked numbers are the coordinates of a vector that is perpendicular to a given plane, that is:
n¯ = [A; B; C]
If the coordinates of three points belonging to the plane are known, then, taking the vector product of two vectors constructed on these points, we can obtain the coordinates n¯. The vector n¯ is called the guide for the plane.
By definition, the dihedral angle formed by the intersection of two planes is equal to the linear angle between their guide vectors. Suppose that we have two planes whose normal vectors are equal:
n 1 ¯ = [A 1 ; B 1 ; C 1 ];
n 2 ¯ = [A 2 ; B 2 ; C 2 ]
To calculate the angle φ between them, you can use the property of the product of the scalar, then the corresponding formula takes the form:
φ = arccos (| (n 1 ¯ * n 2 ¯) | / (| n 1 ¯ | * | n 2 ¯ |))
Or in coordinate 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 )))
We show how to use the described method for calculating dihedral angles in solving geometric problems.
The corners of the regular pyramid of the quadrangular
Suppose that there is a regular pyramid, at the base of which there is a square with a side of 10 cm. The height of the figure is 12 cm. It is necessary to calculate what dihedral angles are equal at the base of the pyramid and for its lateral sides.
Since the figure specified in the condition of the problem is correct, that is, it has high symmetry, then all the angles at the base are equal to each other. The angles formed by the side faces are also the same. To calculate the necessary dihedral angles, we find the direction vectors for the base and two lateral planes. Denote the length of the base side by the letter a, and the height h.
The figure above shows a quadrangular regular pyramid. We write the coordinates of points A, B, C and D in accordance with the introduced coordinate system:
A (a / 2; -a / 2; 0);
B (a / 2; a / 2; 0);
C (-a / 2; a / 2; 0);
D (0; 0; h)
Now we find the direction vectors for the base planes ABC and the two sides ABD and BCD in accordance with the procedure described in paragraph above:
For ABC:
AB¯ = (0; a; 0); AC¯ = (-a; a; 0); n 1 ¯ = [AB¯ * AC¯] = (0; 0; a 2 )
For ABD:
AB¯ = (0; a; 0); AD¯ = (-a / 2; a / 2; h); n 2 ¯ = [AB¯ * AD¯] = (a * h; 0; a 2/2)
For BCD:
BC¯ = (-a; 0; 0); BD¯ = (-a / 2; -a / 2; h); n 3 ¯ = [BC¯ * BD¯] = (0; a * h; a 2/2)
Now it remains to apply the corresponding formula for the angle φ and substitute the side and height values from the problem conditions:
Angle between ABC and ABD:
(n 1 ¯ * n 2 ¯) = a 4/2 ; | n 1 ¯ | = a 2 ; | n 2 ¯ | = a * √ (h 2 + a 2/4);
φ = arccos (a 4/2 / (a 2 * a * √ (h 2 + a 2/4))) = arccos (a / (2 * √ (h 2 + a 2/4))) = 67, 38 o
Angle between ABD and BDC:
(n 2 ¯ * n 3 ¯) = a 4/4; | n 2 ¯ | = a * √ (h 2 + a 2/4); | n 3 ¯ | = a * √ (h 2 + a 2/4);
φ = arccos (a 4 / (4 * a 2 * (h 2 + a 2/4)) = arccos (a 2 / (4 * (h 2 + a 2/4))) = 81.49 o
We calculated the values of the angles that were required to be found by the condition of the problem. The formulas obtained in solving the problem can be used to determine the dihedral angles of quadrangular regular pyramids with any values of a and h.
The angles of a triangular regular pyramid
The figure below shows the pyramid, the base of which is a regular triangle. It is known that the dihedral angle between the sides is straight. It is necessary to calculate the area of the base, if it is known that the height of the figure is 15 cm.
The dihedral angle of 90 o is indicated in the figure as ABC. It is possible to solve the problem using the methodology described above, however, in this case, we will do it easier. We denote the side of the triangle a, the height of the figure is h, the apothem is h b and the side edge is b. Now you can write the following formulas:
S = 1/2 * a * h b ;
b 2 = h b 2 + a 2/4;
b 2 = h 2 + a 2/3
Since the two side triangles in the pyramid are the same, the sides AB and CB are equal and are legs of the triangle ABC. Denote their length x, then:
x = a / √2;
S = 1/2 * b * a / √2
Equating the area of the side triangles and substituting the apothem in the corresponding expression, we have:
1/2 * a * h b = 1/2 / b * a / √2 =>
h b = b / √2;
b 2 = b 2/2 + a 2/4 =>
b = a / √2;
a 2/2 = h 2 + a 2/3 =>
a = h * √6
The area of an equilateral triangle is calculated as follows:
S = √3 / 4 * a 2 = 3 * √3 / 2 * h 2
We substitute the height value from the conditions of the problem, we get the answer: S = 584.567 cm 2 .