Square In A Square: Area Percentage Calculation

Hey guys! Let's dive into a cool geometry problem that involves squares within squares. It might seem a bit complex at first, but trust me, we'll break it down step by step. Our main goal here is to figure out what percentage of the area of a larger square is taken up by a smaller square nestled inside it. We've got some key information: the shaded region (the area between the two squares) has a perimeter of 40 cm and an area of 64 cm². So, grab your thinking caps, and let’s get started!

Understanding the Problem

To really nail this problem, understanding the geometry is key. Imagine a big square with a smaller square sitting perfectly inside it. The shaded region is the space between these two squares. This shaded area's perimeter is the sum of all the sides outlining it, both on the outer and inner edges. Its area is the space covered by this shaded 'frame.' We're given these values (40 cm for the perimeter and 64 cm² for the area), and our mission is to find the percentage of the big square's area that the small square covers.

The challenge lies in connecting the perimeter and area of this shaded region to the side lengths of the squares. Once we know those side lengths, calculating the areas and the final percentage becomes much easier. We need to be strategic in how we use the given information. Remember, the perimeter includes both the inner and outer boundaries of the shaded region, and the area is the difference between the areas of the two squares. By carefully relating these pieces of information, we can form equations and solve for the unknowns. This involves a bit of algebraic manipulation and geometric insight, but that's what makes it fun, right? We're essentially detectives, piecing together clues to solve the puzzle!

Setting Up the Equations

Alright, let's get down to business and translate this geometry into some math! This is where we start building the framework for our solution. The first step is to define our variables. Let's say the side length of the larger square is 'L' and the side length of the smaller square is 'S'. This simple step is crucial because it gives us something concrete to work with. Now, we can express the given information in terms of L and S.

The perimeter of the shaded region is made up of the perimeters of both squares. So, we have the equation: 4L + 4S = 40. Notice that we're adding the perimeters because the shaded region's perimeter includes the outer and inner boundaries. This equation gives us a direct relationship between L and S. The area of the shaded region is the difference between the areas of the two squares, leading to our second equation: L² - S² = 64. This equation relates the squares of the side lengths to the shaded area. Now we have two equations with two unknowns, which is a classic setup for solving a system of equations. Our next step is to use algebraic techniques to solve for L and S. We can simplify the first equation and then use substitution or elimination methods to find the values. This is where our algebraic skills come into play, and it's super satisfying when we start seeing the numbers fall into place!

Solving for Side Lengths

Okay, time to roll up our sleeves and do some algebra magic! We’ve got our two equations: 4L + 4S = 40 and L² - S² = 64. Let’s make the first equation a bit simpler by dividing both sides by 4. This gives us L + S = 10. See how much cleaner that looks? Now, we can use this simplified equation to express one variable in terms of the other. For example, we can write L = 10 - S. This is a crucial step because it allows us to substitute this expression for L into our second equation, which will leave us with just one variable to solve for.

Substituting L = 10 - S into L² - S² = 64, we get (10 - S)² - S² = 64. Now, let's expand that squared term: (100 - 20S + S²) - S² = 64. Notice that the S² terms cancel each other out, leaving us with a much simpler equation: 100 - 20S = 64. Next, we can rearrange the terms to isolate S: 20S = 100 - 64, which simplifies to 20S = 36. Finally, dividing both sides by 20, we find S = 36/20 = 1.8. So, the side length of the smaller square is 1.8 cm. Now that we know S, we can easily find L using the equation L = 10 - S: L = 10 - 1.8 = 8.2. Thus, the side length of the larger square is 8.2 cm. We’ve cracked the code and found our side lengths! Give yourselves a pat on the back, guys. We’re one step closer to the final answer!

Calculating Areas

Alright, now that we've wrestled the side lengths out of those equations, let's move on to calculating the areas. This part is actually pretty straightforward, which is a nice breather after all that algebra! Remember, the area of a square is simply the side length squared. So, for the larger square with side L = 8.2 cm, the area is L² = 8.2² = 67.24 cm². For the smaller square with side S = 1.8 cm, the area is S² = 1.8² = 3.24 cm².

We now have the areas of both squares, which is exactly what we need to find the percentage we're looking for. The problem asks: what percentage of the larger square's area is occupied by the smaller square? To find this, we’ll divide the area of the smaller square by the area of the larger square and then multiply by 100 to express it as a percentage. This is a classic percentage calculation, and it's super useful in all sorts of situations, not just geometry problems. So, let’s set up the calculation: Percentage = (Area of smaller square / Area of larger square) * 100. This formula is the key to unlocking our final answer. It's like the last piece of the puzzle fitting perfectly into place!

Determining the Percentage

Okay, time for the grand finale! We've got all the pieces, now let's put them together and calculate that percentage. We know the area of the smaller square is 3.24 cm² and the area of the larger square is 67.24 cm². So, using our percentage formula, we have: Percentage = (3.24 / 67.24) * 100.

Now, grab your calculators (or your mental math muscles!) and let’s crunch those numbers. Dividing 3.24 by 67.24 gives us approximately 0.0482. Multiplying that by 100, we get about 4.82%. So, the smaller square occupies approximately 4.82% of the area of the larger square. And there you have it, guys! We've successfully solved the problem. We started with a complex geometric scenario, set up equations, solved for side lengths, calculated areas, and finally, found the percentage. This whole process demonstrates how different areas of math connect and how we can use them to solve interesting problems. Give yourselves a huge round of applause! You've earned it.

Conclusion

So, to wrap things up, we've successfully navigated a tricky geometry problem involving squares within squares. We figured out that the smaller square occupies about 4.82% of the larger square's area. We did this by first understanding the problem, setting up equations based on the given perimeter and area of the shaded region, solving for the side lengths of the squares, calculating their areas, and finally, determining the percentage.

This problem is a fantastic example of how different mathematical concepts – geometry, algebra, and percentage calculations – come together. It also shows the importance of breaking down a complex problem into smaller, manageable steps. By defining variables, forming equations, and using systematic algebraic techniques, we were able to unravel the puzzle. Remember, guys, math isn't just about formulas; it's about problem-solving and critical thinking. The skills we've used here can be applied to many other situations, both in math and in real life. Keep practicing, keep exploring, and keep having fun with math! You've got this!