Find the angle between a diagonal of a cube and one of its edges.

54.74°

Step-by-step explanation:

Draw a well labelled cube to have a better representation of the question.

A cube is 3-dimensional, therefore it will have coordinates in x,y and z axis. The diagonal of a cube is from one corner to another corner.

Assuming that cube is unit cube, we need to find n angle at one of its edges will make with the diagonal.

I choose the the diagonal and one edge with the unit vector (1,0,0) and (1,1,1).

a = (1,1,1)

b = (1,0,0)

To calculate the angle between two vectors,

                     a.b = |a||b|cosθ

For simplicity, calculate the dot product, and the magnitudes. Then we will substitute the values of the dot product, and the magnitudes of the vectors to solve for the angle.

Calculating the dot product

a.b = (1,1,1) . (1,0,0)

     = (1 × 1) + (1 × 0) (1 × 0)

     = 1

Calculating the magnitudes

1. Magnitude of (1,1,1)

2. Magnitude of (1,0,0)

Calculating the angle between the two vectors

                 cosθ

                 cosθ

                 cosθ

                 θ

                 θ = 54.7356°

                 θ ≈ 54.74°

so a parallelogram is just like a square base times height. multiply 40 and 39 to get 1560 inches.

a triangle is base times height divided by two. so multiply 7.9 and 5 and then divide it by 2 to get 19.75inches.

the kite can be split into two separate triangles and added together to get the total area. 30 x 4 /2 which equals 60. make sure to double it for the two triangles making 120.