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

[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.

 

[d_leet]

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.

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.

 

[jacy]

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.

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

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

 

[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.

 

[jacy]

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.

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.

 

[HallsofIvy]

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.

 

[jacy]

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.

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.