To find the angle between a diagonal of a cube and one of its edges, we first need to understand the geometric properties involved. A cube has all its edges of equal length, say a. The body diagonal of a cube connects two opposite corners and can be calculated using the 3D distance formula.
The length of the diagonal d is given by:
d = √(a² + a² + a²) = √(3a²) = a√3
Now, let’s consider one edge of the cube aligned along the x-axis, which has a length of a. To find the angle θ between the diagonal and this edge, we use the cosine formula. The cosine of the angle is given by the ratio of the adjacent side (length of the edge) to the hypotenuse (length of the diagonal):
cos(θ) = (length of edge) / (length of diagonal) = a / (a√3) = 1/√3
Now, to find the angle θ, we take the arccosine:
θ = cos-1(1/√3)
Calculating this gives approximately:
θ ≈ 54.74°
Rounding this to the nearest degree, the angle between the diagonal of a cube and one of its edges is 55°.