Finding the Volume of a Cube from its Surface Area

This problem asks us to calculate the volume of a cube given its surface area. To do this, we first need to find the length of one side of the cube using the surface area formula, and then use that side length to calculate the volume using the volume formula.

Understanding Cube Formulas

A cube is a three-dimensional shape with six identical square faces. Let 'a' be the length of one side of the cube.

• The surface area of a cube is the total area of all six faces. Since each face is a square with area \(a^2\), the total surface area (SA) is given by the formula:
• \[SA = 6a^2\]
• The volume of a cube (V) is the amount of space it occupies, given by the formula:
• \[V = a^3\]

Step-by-Step Calculation

We are given that the surface area of the cube is 5046 cm².

Step 1: Calculate the Side Length (a)

We use the surface area formula:

\[SA = 6a^2\]

Substitute the given surface area value:

\[5046 \text{ cm}^2 = 6a^2\]

To find \(a^2\), divide the surface area by 6:

\[a^2 = \frac{5046}{6}\]

\[a^2 = 841\]

Now, take the square root of both sides to find the side length 'a':

\[a = \sqrt{841}\]

\[a = 29 \text{ cm}\]

So, the length of one side of the cube is 29 cm.

Step 2: Calculate the Volume (V)

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

\[V = a^3\]

Substitute the value of 'a' we found:

\[V = (29 \text{ cm})^3\]

\[V = 29 \times 29 \times 29 \text{ cm}^3\]

Let's perform the multiplication:

\[29 \times 29 = 841\]

\[841 \times 29 = 24389\]

So, the volume of the cube is:

\[V = 24389 \text{ cm}^3\]

Conclusion

Given the surface area of the cube is 5046 cm², the side length is 29 cm, and the volume of the cube is 24389 cm³.

Revision Table: Cube Calculations

Additional Information: Properties of a Cube

Cubes are fundamental shapes in geometry. Here are some key properties:

• A cube has 6 faces, 12 edges, and 8 vertices.
• All faces are congruent squares.
• All edges are of equal length.
• At each vertex, three faces meet and three edges meet at right angles.
• The diagonal of a face (d) is given by \(d = a\sqrt{2}\).
• The space diagonal (D) connecting opposite vertices is given by \(D = a\sqrt{3}\).
• Cubes are a special type of cuboid where all edges are equal.