Hexagon Area After Cube Enlargement A Step-by-Step Solution

Hey guys! Let's dive into a cool math problem today. We're going to explore what happens to the area of a hexagon when a cube it's related to gets bigger. Imagine a cube, right? Now picture slicing it in a way that reveals a hexagon. What happens to that hexagon's area if we blow up the cube? Sounds intriguing, doesn't it? Let's break it down step by step. This is the perfect blend of geometry and scaling, and trust me, it’s super interesting once you get the hang of it.

The Cube and the Hexagon

So, we kick things off with a cube. A classic, six-sided shape where all sides are equal. In our case, each side of the cube is 2 cm long. Now, here's the clever part: imagine slicing this cube through its center with a plane that cuts through six of its edges. What shape do you think you'll see where the knife goes through? Bingo! It's a hexagon. Not just any hexagon, though – a regular hexagon, meaning all its sides and angles are equal. This hexagon is formed by connecting the midpoints of those six edges we sliced through. Visualizing this is key, so maybe grab a cube-shaped object or sketch one out to help you see it.

The challenge here lies in figuring out how the dimensions of this hexagon relate to the dimensions of the cube. Since the hexagon's vertices are at the midpoints of the cube's edges, we can use some basic geometry – particularly the Pythagorean theorem – to figure out the side length of the hexagon. Think of the right triangles formed within the cube; the hexagon's sides will be the hypotenuses of these triangles. Once we know the side length, we can calculate the area of the hexagon. Remember, the area of a regular hexagon can be found using a neat little formula that involves the side length. Understanding this initial relationship between the cube and the inscribed hexagon is crucial because it sets the stage for understanding how scaling the cube affects the hexagon's area. It's like the foundation upon which we build the rest of our solution. We need to be crystal clear on this before we can even think about scaling things up. So, let’s make sure we’ve got this part down pat before moving on to the next exciting stage: the enlargement.

Calculating the Initial Hexagon Area

Alright, let’s get down to the nitty-gritty and figure out the actual side length and area of our initial hexagon. Remember, our cube has sides of 2 cm each. The hexagon is formed by connecting the midpoints of the cube's edges. This means that each vertex of the hexagon is exactly halfway along the edge of the cube. Now, imagine a right-angled triangle formed by half of one edge (1 cm), another half of an adjacent edge (also 1 cm), and the side of the hexagon as the hypotenuse. We can use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the length of the hexagon's side. So, $1^2 + 1^2 = c^2$, which simplifies to $2 = c^2$. Taking the square root of both sides, we find that the side length of the hexagon, let's call it 's', is $\sqrt{2}$ cm. Great! We’ve got our side length.

Now, let’s tackle the area. A regular hexagon can be thought of as six equilateral triangles all joined together. The formula for the area of an equilateral triangle is $(\sqrt{3}/4) * a^2$, where 'a' is the side length. Since our hexagon has six of these triangles, we just multiply that area by 6. So, the area of the hexagon is $6 * (\sqrt{3}/4) * s^2$. We already know that $s = \sqrt{2}$, so let’s plug that in: $6 * (\sqrt{3}/4) * (\sqrt{2})^2$. This simplifies to $6 * (\sqrt{3}/4) * 2$, which further simplifies to $3\sqrt{3}$ square centimeters. Boom! That’s the area of our original hexagon. It’s essential to understand this base area because it’s what we'll be scaling up in the next step. If this part isn't crystal clear, it’s worth going back over the steps to make sure you’ve got it. With this foundational knowledge in place, we're ready to explore what happens when we enlarge the cube and, consequently, the hexagon. Let's move on and see how that scale factor affects the area!

Scaling Up: The Enlargement Factor

Okay, guys, here's where it gets even more interesting! We're not just dealing with the initial hexagon anymore; we're about to blow it up – metaphorically, of course. The problem tells us that the cube is enlarged by a scale factor of 3. Now, what does this mean for our hexagon? Well, a scale factor of 3 means that every dimension of the cube is multiplied by 3. So, if the original cube had sides of 2 cm, the enlarged cube will have sides of 2 cm * 3 = 6 cm. It's like we've taken our cube and stretched it out in all directions, making it three times bigger in each dimension.

But how does this scaling of the cube affect the hexagon inside? Think about it: the hexagon is formed by connecting the midpoints of the cube's edges. If the cube's edges get three times longer, then the distances between those midpoints will also increase by a factor of 3. This means that the side length of the enlarged hexagon will be 3 times the side length of the original hexagon. Remember, we calculated the original side length to be $\sqrt{2}$ cm. So, the side length of the enlarged hexagon will be $3 * \sqrt{2}$ cm. See how that scale factor directly impacts the hexagon? It's a beautiful example of how scaling in geometry works.

However, and this is a crucial point, the area doesn't just scale linearly. When we're dealing with areas, we're dealing with two dimensions. So, the area scales by the square of the scale factor. This is a super important concept to grasp. If the length increases by a factor of 3, the area increases by a factor of $3^2$, which is 9. This is because area is a two-dimensional measurement (like square centimeters), and both dimensions are being scaled up. So, we now know that the area of the enlarged hexagon will be 9 times the area of the original hexagon. This understanding of how scale factors affect area is essential for solving this problem and for many other geometry problems. Are you ready to calculate the final area? Let's do it!

Calculating the Enlarged Hexagon Area

Alright, let's bring it all together and calculate the area of our enlarged hexagon. We've already done the hard work of figuring out the scaling factor and how it affects the area. We know that the scale factor is 3, and since area scales by the square of the scale factor, the enlarged hexagon's area will be 9 times the area of the original hexagon. Remember, we calculated the area of the original hexagon to be $3\sqrt{3}$ square centimeters. So, to find the area of the enlarged hexagon, we simply multiply this by 9.

Area of enlarged hexagon = $9 * 3\sqrt{3} = 27\sqrt{3}$ square centimeters. And there you have it! That's the final answer. It's a testament to the power of understanding scale factors and how they affect geometric shapes. This problem beautifully illustrates how a simple enlargement of a cube can have a predictable and calculable effect on a shape inscribed within it.

So, to recap, we started with a cube, sliced it to reveal a hexagon, figured out the area of that hexagon, scaled up the cube, and then calculated the area of the enlarged hexagon. It's a journey through geometry that touches on key concepts like the Pythagorean theorem, area calculations for regular hexagons and equilateral triangles, and, most importantly, the concept of scale factors and their effect on area. I hope you found this explanation clear and helpful. If you ever encounter a similar problem, remember to break it down step by step, visualize the shapes involved, and remember those key formulas and principles. You've got this!

Conclusion

In conclusion, by scaling the cube by a factor of 3, the area of the hexagon formed within it increases by a factor of 9. This demonstrates a fundamental principle in geometry: linear dimensions scale directly with the scale factor, but areas scale with the square of the scale factor. The final area of the enlarged hexagon is $27\sqrt{3}$ square centimeters. This exercise not only reinforces mathematical concepts but also enhances spatial reasoning and problem-solving skills. Remember, geometry is all about seeing shapes and relationships, so keep practicing and exploring!