Calculating Cube Volume From Surface Area A Comprehensive Guide

Determining the volume of a cube when given its surface area is a classic problem in geometry. This article aims to provide a comprehensive explanation of how to solve this type of problem, using the specific example where the surface area of the cube is $24a^2$. We will explore the underlying formulas and step-by-step calculations required to arrive at the correct volume. Understanding these principles is crucial for anyone studying geometry, whether in school or for personal enrichment.

Surface Area of a Cube

To begin, let's discuss the surface area of a cube. A cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. The surface area is the total area of all these faces combined. If we denote the side length of the cube as 's', then the area of one face is $s^2$, since each face is a square. Because there are six faces, the total surface area (SA) of the cube is given by the formula:

$\qquad SA = 6s^2$

In our problem, we are given that the surface area is $24a^2$. Thus, we can set up the equation:

$\qquad 6s^2 = 24a^2$

To find the side length 's', we need to solve this equation for 's'. First, divide both sides by 6:

$\qquad s^2 = \frac{24a^2}{6}$ $\qquad s^2 = 4a^2$

Now, take the square root of both sides:

$\qquad s = \sqrt{4a^2}$ $\qquad s = 2a$

So, the side length of the cube is $2a$. This is a crucial step, as knowing the side length is necessary to calculate the volume. Remember, understanding the relationship between the surface area and the side length is fundamental in solving such geometric problems. We’ve carefully walked through the process of deriving the side length, ensuring that each step is clear and easy to follow. This meticulous approach is essential in mathematics to avoid errors and build a strong foundation.

Volume of a Cube

Now that we have determined the side length of the cube, we can move on to calculating its volume. The volume (V) of a cube is the amount of space it occupies and is calculated by cubing the side length. In other words, the formula for the volume of a cube is:

$\qquad V = s^3$

We found earlier that the side length $s$ is equal to $2a$. Substituting this into the volume formula, we get:

$\qquad V = (2a)^3$

To calculate $(2a)^3$, we need to cube both the number and the variable:

$\qquad V = 2^3 * a^3$ $\qquad V = 8a^3$

Therefore, the volume of the cube is $8a^3$. This is a straightforward calculation once the side length is known, but it's essential to understand the principle behind it. The volume represents the three-dimensional space enclosed by the cube, and cubing the side length gives us this measure. Understanding this concept is vital for various applications, from packing problems in logistics to calculating the capacity of containers in engineering. Furthermore, this step reinforces the importance of accurate algebraic manipulation and understanding the properties of exponents. We have meticulously shown each calculation to ensure clarity and accuracy, which are paramount in mathematical problem-solving.

Detailed Solution

Let's recap the entire solution process step-by-step to ensure a comprehensive understanding. This methodical approach is key to tackling geometry problems effectively. We started with the given surface area and systematically worked our way to finding the volume.

1. Given Surface Area: The surface area of the cube is given as $24a^2$.
2. Surface Area Formula: Recall that the surface area of a cube is $6s^2$, where $s$ is the side length.
3. Set Up the Equation: Set the given surface area equal to the formula: $6s^2 = 24a^2$.
4. Solve for Side Length: Divide both sides by 6 to get $s^2 = 4a^2$. Then, take the square root of both sides to find $s = 2a$.
5. Volume Formula: The volume of a cube is given by $V = s^3$.
6. Substitute Side Length: Substitute the value of $s$ (which is $2a$) into the volume formula: $V = (2a)^3$.
7. Calculate Volume: Simplify $(2a)^3$ to $8a^3$.

Thus, the volume of the cube is $8a^3$. This detailed step-by-step solution not only provides the answer but also illustrates the logical progression required to solve the problem. Each step builds upon the previous one, highlighting the importance of understanding the underlying concepts and formulas. By breaking down the problem into manageable steps, we can avoid confusion and ensure accuracy. This approach is particularly beneficial for students learning geometry, as it encourages a systematic and thorough method of problem-solving. Moreover, this comprehensive explanation serves as an excellent reference for anyone looking to understand or review cube volume calculations.

Conclusion

In summary, we have successfully found the volume of a cube with a surface area of $24a^2$. The process involved using the surface area formula to determine the side length and then applying the volume formula. The crucial steps were:

• Setting up the equation $6s^2 = 24a^2$.
• Solving for the side length $s$ to get $s = 2a$.
• Substituting the side length into the volume formula $V = s^3$.
• Calculating the volume $V = (2a)^3 = 8a^3$.

The correct answer is therefore C) $8a^3$. This exercise demonstrates the importance of understanding basic geometric formulas and algebraic manipulations. Mastering these skills is essential for success in mathematics and related fields. By carefully following the steps outlined in this article, anyone can confidently solve similar problems involving cubes and their surface areas and volumes. This example serves as a valuable tool for learning and reinforcing geometric principles. Furthermore, this process highlights the interconnectedness of different mathematical concepts, such as algebra and geometry, showcasing how they work together to solve problems. The ability to apply these principles is not only beneficial in academic settings but also in practical, real-world situations where spatial reasoning and calculation are required. By understanding these concepts thoroughly, individuals can enhance their problem-solving skills and analytical thinking, which are valuable assets in various aspects of life.

This problem and its solution underscore the fundamental principles of geometry and provide a clear, concise method for calculating the volume of a cube given its surface area. The step-by-step approach ensures that the process is easily understandable, making it an excellent resource for students and anyone interested in mathematics. The emphasis on understanding the underlying formulas and the logical progression of steps makes this a valuable learning tool. Ultimately, the ability to solve problems like this one not only enhances mathematical skills but also fosters critical thinking and problem-solving abilities that are applicable in many areas of life.