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Dihedral Angle: Definition & Calculation

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Dihedral angles show up frequently in the world around us. Through definition and example, we will learn what a dihedral angle is and how to calculate it. After you finish the lesson, you can test your new-found knowledge with a quiz!

Dihedral Angle

Look around the room you are in right now, and observe where one of the walls of the room meets the ceiling. Notice that where this happens, an angle is formed. That angle is called a dihedral angle.

Dihedral Angles
Dihedral Angle 1

A dihedral angle is the angle between two planes. Recall, that a plane is a flat two-dimensional surface. Look at the wall and the ceiling again, and see that these are both flat two-dimensional surfaces, which makes them planes. Therefore, we see that the angle in-between them is an angle between two planes, so it is a dihedral angle. That's a pretty fancy way of describing a wall, ceiling, and the angle in between them, huh?

Dihedral angles show up anywhere that two planes intersect. For example, a polyhedron is a three-dimensional object with polygons as sides. Polyhedrons have dihedral angles between each of their sides. This is because the sides of a polyhedron are planes, so the angles between them are dihedral angles. For instance, consider a cube.

Dihedral Angles in a Cube
Dihedral Angle 2

We see that each of the sides of the cube is a plane and the angles in-between each of these planes is 90 degrees. Thus, the dihedral angles of a cube are each 90 degrees.

Calculating the Dihedral Angle Between Two Planes

Now that we know what dihedral angles are, we get to the fun part - calculating them! All planes have an equation that identifies them. The equation of a plane takes on this form:

Ax + By + Cz + D = 0

Where A, B, C, and D are constants and x, y, and z are variables. For example, the equation x + 7y - 3z + 8 = 0 is the equation of a plane.

When we know the equations of two planes, Ax + By + Cz + D = 0 and Ex + Fy + Gz + H = 0, we can use the following formula to find the angle between the two planes.

Dihedral Angle Formula
Dihedral Angle 3

To find the dihedral angle between two planes, we do the following.

Put your plane equations in this form

Ax + By + Cz + D = 0

Ex + Fy + Gz + H = 0

Identify your A, B, C, E, F, and G.

2. Plug these values into your formula.

3. Simplify as much as possible, then use a calculator to find the angle.

For example, consider these two planes:

x + 7y - 3z + 8 = 0

3x - 2y + 4z - 1 = 0

These equations are already in the correct form, so to find the angle between these two plane, we first identify A, B, C, E, F, and G.

A = 1

B = 7

C = -3

E = 3

F = -2

G = 4

We plug these into our formula to get the following:

Dihedral Angle 4

The dihedral angle between the two planes is 123.782 degrees (rounded to three decimal places).

Example Problem

Suppose you want to hang a ceiling fan in the corner of a vaulted ceiling. The ceiling fan's hanger has an angle of 120 degrees. The two sides of the ceiling are planes with the following equations:

2x + 8y - 2z - 4 = 0

2x - y + 5z + 4 = 0

Will the ceiling fan hanger fit in the corner of the vaulted ceiling?

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