Calculating Cube Volume from Surface Area

The problem asks us to find the volume of a cube given its total surface area. We are provided with the total surface area as 24 square units.

To solve this, we need to use the formulas for the surface area and volume of a cube.

Cube Formulas

• The total surface area of a cube is given by the formula: Total Surface Area $= 6 \times (\text{side})^2$. Let 's' represent the length of one side of the cube. So, Total Surface Area $= 6s^2$.
• The volume of a cube is given by the formula: Volume $= (\text{side})^3$. So, Volume $= s^3$.

Step-by-Step Calculation

Step 1: Find the side length of the cube.

We know the total surface area is 24 sq. units. Using the surface area formula:

$\text{Total Surface Area} = 6s^2$

$24 = 6s^2$

To find $s^2$, we divide the total surface area by 6:

$s^2 = \frac{24}{6}$

$s^2 = 4$

Now, to find the side length 's', we take the square root of 4:

$s = \sqrt{4}$

Since the side length must be a positive value:

$s = 2$ units

So, the length of each side of the cube is 2 units.

Step 2: Calculate the volume of the cube.

Now that we have the side length, we can use the volume formula:

$\text{Volume} = s^3$

Substitute the value of 's' we found:

$\text{Volume} = (2)^3$

$\text{Volume} = 2 \times 2 \times 2$

$\text{Volume} = 8$ cubic units

Thus, the volume of the cube is 8 cubic units.

Let's verify with the options provided:

Our calculated volume matches the first option.

Revision Table: Cube Properties

Additional Information: Understanding Cube Dimensions

A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meetings at each vertex. It is a type of hexahedron.

• All edges of a cube are of equal length.
• All faces of a cube are congruent squares.
• A cube has 6 faces, 12 edges, and 8 vertices.
• The calculation of surface area involves finding the area of one square face ($s^2$) and multiplying it by the number of faces (6).
• The calculation of volume involves multiplying the length, width, and height of the cube. Since all are equal to the side length 's', the volume is $s \times s \times s = s^3$.

Problems involving cubes often require you to use these fundamental formulas to find unknown dimensions or properties based on known values.