How to calculate the volume of a cube?

The cube is a polyhedron with six faceseach in the shape of square and with the same dimensions, joined together by right angles, which is why it is also known as regular hexahedron. The easiest way to calculate its volume is cube or third power the length of one of its edges.

Calculate the volume knowing the length of an edge

The volume of a regular polyhedron is calculated by multiplying the height, width, and depth. Since all faces of the cube are square, the height, width, and depth are equal. Thus, raising to the cube, or third power, the length of one of its edges, the volume of the cube would be obtained:

Basic formula for the volume of a cube

Where:

  • v is volume
  • a is the length of one of the edges.

For example, if an edge measures 8 m, its volume would be 512 m3:

It is important to write down the units correctly. The volume measures a three-dimensional space and therefore volume is expressed in cubic units. For example, if we have the length of the edge in cm, the result will be expressed in cm3 (cubic centimeters), or if we had the length in m, the result would be expressed in m3 (cubic meters).

There are specific units of volume, such as the Liter, all of which are interchangeable with cubic units. For example, 1 L (liter) is equal to 1 dm3 (cubic decimeter) and 1 ml (milliliter) is equal to 1 cm3.

Get the volume knowing the surface of the cube

The cube has six square faces. Therefore, if we know the total area of ​​the cube and we split it between 6we will get the surface of one of its faces. From this surface we can calculate the length of an edge and calculate the volume in the same way as with the previous method.

Let us imagine that the total surface of the cube is 150 cmtwothe surface s of each of the faces would be:

s = 150/6 = 25 cm2

So, we have that one face of the cube has an area of ​​25 cmtwo. Since the face is square and the area of ​​a square is calculated by multiplying the length of one side by itself, that is, raising the length of one side to the second power (atwo), then we can calculate how long one side is by doing the square root of its area:

a = √25 = 5 cm

Since we already have the length of one edge of the cube, we can now calculate its volume raising this value to the third power:

V = 53 = 125 cm3

Summarizing, if we know the total surface of the cube:

  1. We divide the surface of the cube by 6 to obtain the surface of one face
  2. We calculate the square root of the surface of a face to obtain the length of a side
  3. We raise the length of one side to the cube and we already have the volume

Obtain the volume knowing a diagonal

If a diagonal is drawn on one of the faces of the cube, a right triangle is obtained to which the Pythagoras theorem.

Calculate volume of cube with diagonals

the diagonal d would be the hypotenuse of the triangle, so the diagonal squared would be equal to one edge of the cube squared plus another edge of the cube squared:

d2 = a2 + a2

Since the edges in the cube measure the same:

d2 = 2a2

Solving, we obtain that this diagonal is equal to √2 times the length of a side, that is:

d = a√2

So we can divide the length of the diagonal by √2 to find the length of the edge. We raise it to the cube and we will obtain the volume of the cube:

a = d/√2
V = a3

If instead of the diagonal of a face, we know the three-dimensional diagonal from one vertex to its opposite, Dand we apply the Pythagorean Theorem:

D2 = d2 + a2

Earlier we calculated that dtwo = 2atwoso:

D2 = 3a2

Solving, the length of the edge would be equal to the diagonal D between √3:

a = D/√3
V = a3

Namely:

  1. if we know the diagonal of a face, we divide the diagonal by √2
  2. if we know the three-dimensional diagonal from one vertex to the opposite, we divide the diagonal by √3;

The result is the length of one side, we raise it to the cube and we can now calculate the volume of the cube.

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