The angle between two face diagonals that share a common vertex in a cube is 60 degrees. This fundamental property of a cube arises from the symmetrical nature of its structure.
Understanding Face Diagonals
A face diagonal is a line segment connecting two opposite vertices on a single face of a three-dimensional shape. In a cube, each of its six square faces has two such diagonals.
- Length of a Face Diagonal: For a cube with side length s, the length of a face diagonal can be calculated using the Pythagorean theorem. If we consider one face, the diagonal acts as the hypotenuse of a right-angled triangle formed by two adjacent edges.
- Length² = s² + s²
- Length² = 2s²
- Length = s√2
This means every face diagonal in a cube has the same length, which is s√2.
Deriving the Angle: A Geometrical Approach
To determine the angle between two face diagonals that share a common vertex, let's consider a cube with side length s.
- Identify a Common Vertex: Choose any corner of the cube. Let's call this vertex 'A'.
- Select Two Adjacent Faces: From vertex 'A', three faces meet. Pick any two of these adjacent faces. For instance, consider the top face and a side face.
- Draw the Face Diagonals:
- On the top face, draw a diagonal starting from 'A' to the opposite corner of that face. Let's call this diagonal AD1. Its length is s√2.
- On the adjacent side face, draw another diagonal also starting from 'A' to the opposite corner of that specific side face. Let's call this diagonal AD2. Its length is also s√2.
- Form an Equilateral Triangle: Now, consider the line segment connecting the endpoints of AD1 and AD2 (the corners opposite to 'A' on their respective faces). This segment is also a face diagonal on the third face of the cube. Therefore, its length is likewise s√2.
Since all three sides of the triangle formed by AD1, AD2, and the connecting face diagonal are equal in length (s√2), this triangle is an equilateral triangle. All angles in an equilateral triangle are 60 degrees. Thus, the angle between the two face diagonals (AD1 and AD2) is 60 degrees.
Key Properties of a Cube and its Diagonals
Understanding the various types of diagonals within a cube helps to contextualize this specific angle.
| Feature | Description | Length (for side s) |
|---|---|---|
| Edge | A line segment connecting two vertices. | s |
| Face Diagonal | Connects opposite vertices on a single face. | s√2 |
| Body Diagonal | Connects opposite vertices through the cube's interior. | s√3 |
Other Angles in a Cube
While the angle between two adjacent face diagonals is 60 degrees, a cube presents other interesting angles:
- Angle between an edge and a face diagonal on the same face: 45 degrees.
- Angle between a face diagonal and a body diagonal: cos⁻¹(√6/3) ≈ 35.26 degrees.
- Angle between an edge and a body diagonal: cos⁻¹(1/√3) ≈ 54.74 degrees.
Conclusion
The angle between any two face diagonals that originate from the same vertex in a cube is precisely 60 degrees. This property is a direct consequence of the cube's symmetrical geometry, forming an equilateral triangle with sides equal to the length of a face diagonal.
