What is the volume of the cube if the diagonal is 1?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The diagonal of a cube is given as sqrt (3L^2), where L is the length of the side.

We know that the diagonal is 1.

=> 1 = sqrt (3L^2)

=> 1^2 = 3L^2

=> L^2 = 1/3

=> L = sqrt( 1/3)

The volume of a cube is L^3

=> [sqrt (1/3)]^3

=> (1/3) sqrt (1/3)

The required volume of the cube is (1/3)*sqrt (1/3).

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

To find the volume of the cube we need to find the length of the side.

Given the diagonal of the cube is 1.

Let the side of the cube be x.

Then we know that the diagonal is given by:

d = sqrt(3x^2) = (sqrt3)*x

==> 1= sqrt3*x

==> x=  1/ sqrt3

==> x= sqrt3/3

Now we will calculate the volume of the cube.

We know that the volume of the cube = x^3

==> V = (sqrt3/3)^3 = 3sqrt3/ 27 = sqrt3/9

Then, the volume of the cube is sqrt3/ 9 cubic units.

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

A cube is a regular prism which has  a square top, bottom and lateral surfaces with equal length, width and height.

If the side of the cube is x, then its diagonal d is given by:

d = x^2+x^2+x^2 = 3x^2.

So , if diagonal d= 1, then 1= 3x^2 => x= sqrt(1/3) is the side of the cube.

So the volume of the cube = x^3 = (sqrt(1/3)}^3 = (1/3^3/2)= 1/sqrt27 = 0.19245 cubic units.

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