Question

The square on the diagonal of a cube has are 147 cm². Find the length of the side and surface area of the cube.

Answer 1

1. Understanding the relationship between the diagonal and the side:

The diagonal of a cube is the line segment connecting opposite corners of the cube. If the length of the side of the cube is denoted by s, then the diagonal d can be expressed using the Pythagorean theorem in three dimensions:

`d = sqrt(s^2 + s^2 + s^2)`

= `sqrt(3s^2)`

= `ssqrt(3)`

2. Square on the diagonal:

The area of the square on the diagonal is given as 147 cm². Since the diagonal of the square is the diagonal of the cube, the side length of the square will be equal to the diagonal of the cube d.

The area of the square is also related to its side length, say `s_("square")`, by the formula:

Area of square = `s_("square")^2`

And since the diagonal of the square is the diagonal of the cube:

`s_("square") = ssqrt(3)`

Therefore, the area of the square is:

`(ssqrt(3))^2 = 147`

`3s^2 = 147`

`s^2 = 147/3 = 49`

`s = sqrt(49) = 7` cm

3. Surface area of the cube:

The surface area of a cube is given by the formula:

Surface area = 6s²

Substituting s = 7 cm:

Surface area = 6(7)² = 6(49) = 294 cm²