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

In summary: Thanks, so the length of the diagonal will be z2=3y2Then the area will be 18y2 square units. Please correct me if am wrong, thank you.
  • #1
jacy
76
0
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.
 
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  • #2
jacy said:
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.
 
  • #3
d_leet said:
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
 
  • #4
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
StatusX said:
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.
 
  • #6
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 [itex]x= \sqrt{2}y[/itex] 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 [itex]\sqrt{2}y[/itex]. If you let z be the length of that diagonal, z2= y2+ 2y2.
 
  • #7
HallsofIvy said:
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 [itex]x= \sqrt{2}y[/itex] 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 [itex]\sqrt{2}y[/itex]. 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.
 

What is the formula for finding the edge length of a cube?

The formula for finding the edge length of a cube is:
Edge length = Diagonal length / √3

Can you provide an example of using the formula?

For example, if the diagonal length of a cube is 10 inches, the edge length would be:
Edge length = 10 inches / √3 = 10 inches / 1.732 = 5.773 inches

How is the formula derived?

The formula is derived using the Pythagorean theorem. In a cube, the diagonal length forms a right triangle with the edge length as the hypotenuse. Therefore, the formula is:
Edge length = Diagonal length / √3 = √(Edge length² + Edge length² + Edge length²) / √3 = √(3*Edge length²) / √3 = √(Edge length²) = Edge length

Is the formula applicable to all cubes?

Yes, the formula is applicable to all cubes because the relationship between the edge length and diagonal length is constant in all cubes. Therefore, the formula can be used to find the edge length of any cube as long as the diagonal length is known.

Why is it important to express the edge length as a function of the diagonal?

It is important to express the edge length as a function of the diagonal because it allows for a more efficient and accurate calculation. Instead of measuring the edge length directly, which can be difficult in some cases, the diagonal length can be measured and plugged into the formula to find the edge length. This also allows for the calculation of the edge length for cubes with non-integer diagonal lengths.

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