SOLUTION: Show the diagonal of a cube of edge e is square root of 3 multiply by e.

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Question 1194978: Show the diagonal of a cube of edge e is square root of 3 multiply by e.

Found 2 solutions by math_helper, math_tutor2020:
Answer by math_helper(2461) (Show Source):
You can put this solution on YOUR website!

Pick two corners, at opposite ends of a diagonal on any one face:
Diagnoal length of one side:
Now take that diagonal and make a right triangle using any point from the
opposite face (i.e. the face that shares no verticies with the face chosen above). The hypotenuse of this triangle coincides with one of the four (main) diagonals of the cube (which all have the same length), and has length:

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Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

Here's how to set up the diagram of what's going on.
After the diagram is set up, refer to the steps mentioned by the tutor @math_helper

Draw a square ABCD. It can be any size you want.
Then draw a diagonal from A to C (shown in red). This is shown in figure 1.

Now form a cube with ABCD as the floor

The red line AC is entirely along the floor.

Technically it could be any of the 6 faces, but I think the floor is the easiest to go with.
Imagine this cube as a giant room that a person can go into.
For me, this line of thinking helps keep a consistent orientation.

Lastly, draw a segment from A to G
This segment is in blue (see figures 3, 4 and 5)

Figures 3, 4 and 5 are the same cube but from different angles to give you an idea how it would look like if animated to spin around.
The red segment is entirely on the surface of one face only. That face being the floor panel.
The blue segment is inside the cube itself. It is referred to as the space diagonal. In contrast, the red diagonal is a face diagonal.

Face diagonal = diagonal that's on a face only
Space diagonal = diagonal that cuts through the entire cube.

Put another way: The blue segment AG is like a rope connected from the bottom corner on one side of the room, to extend to the upper opposite corner. Stretch the rope out as far as you can. The middle pieces of the blue rope cannot touch the walls, ceiling or floor.

Notes:
Triangle ABC is a right triangle.
Triangle ACG is a right triangle.
Because we have right triangles, we can use the pythagorean theorem to find each diagonal (given we know the side length of the cube).
Each image was created with GeoGebra.
The red line (AC) is shown to be units in length by the other tutor.
The blue line (AG) is exactly units long.