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NY Regents Exam - Geometry: Tutoring Solution10 chapters | 120 lessons

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!

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**.

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.

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.

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* + 7*y* - 3*z* + 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.

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* + 7*y* - 3*z* + 8 = 0

3*x* - 2*y* + 4*z* - 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:

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

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:

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

2*x* - *y* + 5*z* + 4 = 0

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

To see if the ceiling fan hanger will fit, we will calculate the dihedral angle between the two sides of the ceiling and see if the angle is large enough to fit the angle of the ceiling fan hanger in it. Both equations of our planes are in the correct form, so we start by identifying *A*, *B*, *C*, *E*, *F*, and *G*.

*A* = 2

*B* = 8

*C* = -2

*E* = 2

*F* = -1

*G* = 5

Plugging these values into our formula gives the following:

The dihedral angle between the two sides of the ceiling is 107.532 degrees (rounding to three decimal places). Since the ceiling fan hanger is 120 degrees and the dihedral angle between the two sides of the ceiling is only 107.532 degrees, we see that it will not fit. Looks like we have to head back to the hardware store and get a new hanger!

A **dihedral angle** is an angle between two planes, where a **plane** is a flat two-dimensional surface. Anywhere that two planes intersect, there is a dihedral angle. When we know the equations of two planes - *Ax* + *By* + *Cz* + *D* = 0 and *Ex* + *Fy* + *Gz* + *H* = 0 - we can find the dihedral angle between them using the following formula:

We can add this formula to our toolbox along with all of this invaluable information about dihedral angles. This will allow us to be comfortable working with these types of angles in the world around us.

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NY Regents Exam - Geometry: Tutoring Solution10 chapters | 120 lessons

- Triangles: Definition and Properties 4:30
- Area of Triangles and Rectangles 5:43
- Classifying Triangles by Angles and Sides 5:44
- Perimeter of Triangles and Rectangles 8:54
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- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Applications of Similar Triangles 6:23
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
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