How would you write the formula for the surface area and volume of a cube in terms of the length of its diagonal?

Question

I am given the variable d which represents the diagonal of the cube. There are no other numbers or variables given. I must write the formula to find the surface area of the cube

Answer 1

The ratio of a diagonal to one of the sides is sqrt 3.

so to find the volume, take the length of the diagonal and divide by sqrt 3 to find the length of one side

use the normal formula to find the volume given that side.

Answer 2

A cube consists of congruent sides and each face of a cube is a square.

The area of a square is a² and because the cube has six sides, the surface area of the cube is 6 (a² ) or 6a²

A diagonal on the face of the cube will create two 45-45-90 triangles where the legs = x and the hypotenuse = x√2.

The diagonal d is also the hypotenuse

d=x√2

Solving for x will yield the length of each side .

d/√2=x√2/√2

x = d/√2

x= (d)(√2) / (√2)(√2) = d√2 / 2

To calculate the area of one side, determine x²

(d√2 / 2)² = 2d² / 4 = (1/2)d²

The area of 6 side is (6)(1/2)d² = 3d² which is the surface area.

SA = 3d²

The volume is a³ or x³ = (d)(√2) / 2 * (d)(√2) / 2 * . (d)(√2) / 2 . = d³ * . 2√2 /8 =

V = d³√ 2 /4

Volume is d cubed multiplied by the square root of 2 divided by 4