Dihedral Angle
Introduction
Generally, an angle occurs when two lines or line segments intersect each other. And this may be either acute or obtuse or right angle. But now we are discussing Dihedral Angle, which is also an intersection point but between two planes.
Thus, the Dihedral Angle may be defined as an angle that occurs when two planes can intersect each other directly or indirectly. These planes are termed as Cartesian planes or coordinates, which help to determine the shape of objects in two dimensions or threedimension. So let’s discuss more of this, which includes the formula, calculation method, examples, uses, etc.
The Formula for Calculating Dihedral Angle
We need to calculate the dihedral Angle when two Cartesian coordinates or planes intersect each other. Now we need to derive a formula from the vectors of given planes. If an equation may represent the vectors of a plane,
Say, ax+by+cz+d =0.
Here, the vector is denoted as n. And,
n =(a,b,c).
In the same way, we will take vectors for both the planes and the notations can be taken as n1,n2.
So, normal vectors can be written as ,
n1 =(a1,b1,c1) and
n2 =(a2,b2,c2)
Let say 𝚹 will be the dihedral Angle, then the formula can be written as
Cos𝚹 = n1^n2
i.e.,
Cos𝚹 = \[\frac{(n1*n2)}{\sqrt{n1}*\sqrt{n2}}\]
Cos𝚹 = \[\frac{a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}\]
This is known as the formula for the dihedral Angle.
Procedure to Calculate the Dihedral Angle Using this Formula
We need to calculate the dihedral angle, which is the intersection of two planes in geometry either in two dimensional or in three dimensional. For this, we need to follow some sequential steps as given below –

In the first step, we need to determine the values from the figure and represent them in an equation.

Next, we need to denote normal vectors.

Now, calculate the values of the normal vectors.

Finally, substitute all these values in the Dihedral Angle formula.

Then we get the value of the angle between those intersecting planes.
This is the simple procedure we need to follow to calculate the Dihedral Angle. We can understand more clearly by solving certain examples.
Examples to Find Dihedral Angle:
Q. If the planes have equations as, 3x+y+4z =0 and x+4y+z =0. Find the intersecting angle between the planes.
Sol. Given planes are written as,
Plane 1, 3x+y+4z =0. And
Plane 2, x+4y+z = 0.
By comparing these equations with standard notation, we can take the values as,
p1= 3, q1= 1,r1 = 4. and
p2=1, q2= 4,r2= 1.
Then we need to substitute these values in the formula,
Cos𝚹 = \[\frac{(3*1) + (1*4) + (4*1)}{\sqrt{(3*3) + (1*1) + (4*4)}\sqrt{(1*1) + (4*4) + (1*1)}}\]
= \[\frac{(3+4+4)}{\sqrt{(9 + 1 + 16)}\sqrt{(1 + 16 + 1)}}\]
= \[\frac{11}{\sqrt{26}\sqrt{18}}\]
= \[\frac{11}{\sqrt{468}}\]
= 0.50.
Hence, this is the dihedral angle between the given two planes.
Similarly, we can calculate the values of the dihedral angle between different planes.
Scope of Dihedral Angle

Dihedral Angle plays a significant role in mathematics as well as chemistry that, too, in calculating the analysis of protein. It is also helpful in various experiments.

The Dihedral Angle helps to find out the interior angle in polyhedron and tetrahedron.

This Angle plays a vital role in proving the planes are moving parallelly.

If the angle is zero, then the planes are parallel to each other.

Dihedral Angle is either acute or may obtuse based on the intersection point.
Conclusion
Thus, the Dihedral Angle can be defined as an angle that lies between the intersection of two Cartesian coordinates. This Angle helps to solve sums, especially in geometry, which occurs very rarely. The notation, formula, and calculation are simple and easy to understand.
The value of angle also helps in various analyses of chemistry. It has a wide scope with various applications. This is a scoring concept for students and experimental tools for mathematicians and science scholars too. As it is a simple formula to understand and use, everyone can practice it perfectly and achieve your target, which is either score or result or value.
1: What do you Mean by Dihedral Angle?
Ans: Dihedral Angle is an angle which can be defined as the angle occurred when two different planes intersect each other either directly or indirectly.
If the value of the angle shows that it is acute, then we can obtain its obtuse angle value by reducing it from the sum of supplementary angles, which equals 180.’
2: What is the Significance of the Dihedral Angle?
Ans: Dihedral Angle plays a predominant role in mathematics to solve complex equations in geometry and to find the relationship between two different planes. It also helps to determine the shape of objects and provides stability too.
3: What do you Call Intersecting Planes?
Ans: In the dihedral angle, the two planes are called Cartesian planes or coordinates.
4: Does it have any Formula to Find the Angle?
Ans: Yes, it has a simple formula using vectors that is there to find the dihedral angle. This can be taken from the vectors of the planes. Then we will take a normal value from those vectors and substitute all the values in the formula.
5: Is it Possible to Prove that the Planes are Parallel Using this Dihedral Angle?
Ans: Yes, of course, it is very easy to prove that the planes are parallel to each other. If the planes are parallel, then there is no intersection point between them. Thus, if the dihedral angle is zero, the planes are parallel to each other.
6: Differentiate Dihedral Angle and Torsion Angle.
Ans: Both the angles deal with planes itself, but the difference occurs in finding perspective. Dihedral Angle finds the value of the intersection point, whereas the torsion angle finds out the direction of planes, whether it is positive or negative. But both the angles have their own applications and are reliable too.