A diagonal of a cube is a line segment of maximum length whose endpoints are vertices of the cube. To the nearest degree, what is the angle of intersection Ѳ,
0 < Ѳ <= 90° between two distinct diagonals of the same cube?
This problem is discussed at
https://answers.yahoo.com/question/index?qid=20101127042533AA17HTY
To find the angle of intersection between two distinct diagonals of the same cube, we can follow these steps:
Step 1: Understand the cube:
- A cube has 6 equal square faces.
- The length of each side of the cube is denoted as "s".
Step 2: Determine the length of the diagonals:
- Since each face of the cube is a square, the length of each diagonal of the square face can be found using the Pythagorean theorem.
- The length of the diagonal of a square face is "s√2".
Step 3: Identify the diagonals that intersect:
- A cube has two types of diagonals: face diagonals and space diagonals.
- Face diagonals connect opposite corners of the same face.
- Space diagonals connect opposite corners of the cube.
Step 4: Find the angle of intersection:
- The diagonal within each face of the cube intersects with the diagonals of the adjacent faces at a right angle (90°).
Let's focus on the space diagonals since the question asks for the angle between two distinct diagonals.
Step 5: Identify the space diagonals:
- There are four space diagonals in a cube.
- Each space diagonal connects the opposite corners of the cube:
- Diagonal 1: Connects two corners of one face to two opposite corners of an adjacent face.
- Diagonal 2: Connects two corners of another face to two opposite corners of an adjacent face.
- Diagonal 3: Connects two remaining corners of the cube.
- Diagonal 4: Connects the remaining two corners of the cube.
Step 6: Calculate the angle of intersection:
- Since all adjacent faces of a cube are perpendicular to each other, the angle between any two diagonals is 90°.
- Therefore, the angle of intersection between two distinct space diagonals of the same cube is 90°.
Therefore, the angle of intersection (Ѳ) between two distinct diagonals of the same cube is approximately 90°.