Express the edge length of a cube as a function of the cube's diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length.

Scratch all that, since

s = d/√3
v = s^3 = d^3/√27
a = 6s^2 = 2d^2

d^2 = 3s^2

so, s = (d/3)^(1/2)
v = s^3 = (d/3)^(3/2)
A = 6s^2 = 2d

edge length s

diagonal of a side = s sqrt 2
then right triangle with legs s and s sqrt 2
d^2 = s^2 + 2 s^2 = 3 s^2
d = s sqrt 3

s = d/ sqrt 3 = d (3)^-.5

area = 6s^2 = 6d^2 (3)-1 = 6d^2/3 =2 d^2

vol=s^3=[d(3^-.5)]^3 = d^3 /(3 sqrt 3)
= (d^3 sqrt 3)/9

whew :)

thank you :)

Well, well, well, let's dive into the world of cubes, shall we?


Now, when it comes to expressing the edge length of a cube as a function of its diagonal length, it's quite simple! We just need a little bit of Pythagorean magic here. You see, in a cube, the edge, diagonal, and a face diagonal form a right triangle.

So, by unleashing the power of Pythagoras, we can find that the edge length (let's call it "e") of the cube can be expressed as:

e = √(d^2/3)

Impressive, right? We've got our edge length all squared away!

Now, let's move on to the surface area and volume. The surface area (S) is basically the sum of the areas of all six faces of the cube. Since all the faces of a cube are equal and congruent, we can express the surface area as:

S = 6e^2

And last but not least, the volume (V) of the cube can be calculated by simply cubing the edge length:

V = e^3

There you have it! We've managed to express the edge length, surface area, and volume of a cube in terms of the diagonal length. Math has never been this entertaining, has it?

To express the edge length of a cube as a function of the diagonal length, we can use the Pythagorean theorem. In a cube, the diagonal of a face is equal to the hypotenuse of a right triangle formed by one edge of the cube and the diagonal.

Let's consider a cube with edge length "a." The diagonal length "d" is the hypotenuse of the right triangle with two sides of length "a" each. So, we can write:

a^2 + a^2 = d^2.

Simplifying this equation, we get:

2a^2 = d^2.

Hence, we can express the edge length "a" as a function of the diagonal length "d" by taking the square root of both sides:

a = sqrt(d^2/2).

Now, let's express the surface area of the cube as a function of the diagonal length "d." The surface area of a cube is given by 6 times the square of the edge length:

Surface Area = 6a^2.

Substituting the value of "a" we obtained earlier, we have:

Surface Area = 6(d^2/2).

Simplifying this expression, we get:

Surface Area = 3d^2.

Finally, let's express the volume of the cube as a function of the diagonal length "d." The volume of a cube is given by the cube of the edge length:

Volume = a^3.

Substituting the value of "a" we obtained earlier, we have:

Volume = (d^2/2)^3.

Simplifying this expression, we get:

Volume = (d^6)/8.