# Express the edge length of a cube as a function of the cubes diagonal

1. Jan 24, 2006

### jacy

Hello,
Here is my what am trying to solve.
Express the edge length of a cube as a function of the cubes diagonal. Then express the area as a function of diagonal length if the side is x.

This is what i know. The area of a cube is 6x^2 where x is the length. But in the problem i have to express the side length as a function of cubes diagonal. Can someone suggest me the relation between the side length and the diagonal, thanks.

Last edited by a moderator: Jan 7, 2014
2. Jan 24, 2006

### d_leet

Think abut the pythagorean theorem and what is true about cubes but not about all rectangular prisms.

And I'm not entirely sure from your post whether you want a formula for surface area or volume.

3. Jan 25, 2006

### jacy

Okay on using the pythagorean theorem am getting
y=2x^2 where y is the diagonal lenght and x is the side length.

Now the area of the cube = 12x^4
am i right. Plz let me know

4. Jan 25, 2006

### StatusX

You gave the corret area formula in your first post. And remember that the pythagorean theorem is a2+b2=c2. You forgot to square c in your formula. And are you interested in the diagonal of a face or of the whole cube? Because they have different lengths.

5. Jan 26, 2006

### jacy

Thanks for correcting me. Now i have y=sqrt 2x

the area of the cube will be 24 x2
The question says "express the area as a function of diagonal length", so am not sure whether to consider the diagonal of a face or the whole cube. 24x2 will be the answer right.

6. Jan 26, 2006

### HallsofIvy

Staff Emeritus
Be careful- the "diagonal" of a cube is NOT the diagonal across a face. It is the line from one vertex of the cube to the opposite vertex. Yes, if y is the length of a side, then $x= \sqrt{2}y$ is the length of the diagonal of a face of the cube. Now, the diagonal from one vertex of a cube to its opposite vertex is the hypotenuse of a right triangle, having one side of length y and the other $\sqrt{2}y$. If you let z be the length of that diagonal, z2= y2+ 2y2.

7. Jan 26, 2006

### jacy

Thanks, so the length of the diagonal will be z2=3y2
Then the area will be 18y2 square units. Please correct me if am wrong, thank you.