How Much Fabric Henri Needs For His Square Pyramid Lantern

Hey guys! Let's dive into a fun geometry problem where Henri is building a cool lantern in the shape of a square pyramid. He's planning to cover the entire pyramid with fabric, and we need to figure out just how much fabric he'll need. Think of it like wrapping a present, but the present is a pyramid! So, let’s break down the problem step-by-step and make sure we nail the solution. We’ve got some answer options lined up: A. $16 m^2$, B. $24 m^2$, C. $36 m^2$, and D. $40 m^2$. Let’s find out which one is the right fit!

Understanding the Square Pyramid

First things first, let's really get what a square pyramid is. Imagine a square as the base – that's the bottom part. Now, picture four triangles, each perfectly identical, rising up from each side of the square and meeting at a single point at the top. That's your square pyramid! It's super important to visualize this because the amount of fabric Henri needs will depend on the total surface area of all these shapes combined. We're not just talking about the square base here; we need to consider all four triangular faces as well. To calculate the total fabric, we need to find the area of the square base and the area of each triangle, then add them all up. Think of it like this: if you were to unfold the pyramid, you’d see one square and four triangles laid out flat. Each of these shapes contributes to the total surface area, and understanding their individual areas is key to solving our problem. We'll need to remember the formulas for the area of a square (side * side) and the area of a triangle (1/2 * base * height). Let's keep these in mind as we move forward, ensuring we have all the tools ready to tackle this geometry challenge!

Calculating the Area of the Base

Okay, let's start with the foundation of Henri's lantern – the square base. To figure out how much fabric we need for this part, we've got to calculate the area of the square. Remember, the area of a square is simply the length of one side multiplied by itself. In our case, the problem states that the base has sides of 4 meters each. This means we're looking at a square that's 4 meters wide and 4 meters long. So, to find the area, we do a quick calculation: 4 meters * 4 meters. That gives us 16 square meters. Easy peasy, right? This tells us that Henri will need 16 square meters of fabric just to cover the bottom of the lantern. But we're not done yet! We still have those four triangular faces to think about. The base is just one part of the whole pyramid, and we need to account for all the sides to get the total amount of fabric required. So, let's keep this 16 square meters in our minds and move on to calculating the area of those triangles. We’re one step closer to figuring out the grand total!

Determining the Area of One Triangular Face

Alright, let's tackle the triangles! Each triangular face of the pyramid plays a crucial role in the overall surface area. Remember, the formula for the area of a triangle is 1/2 * base * height. The base of each triangle is the same as the side of the square base, which we know is 4 meters. Now, here's where it gets a little tricky – we need the height of the triangle. The problem tells us that the triangular faces have a height of 5 meters. This is the perpendicular distance from the base of the triangle to its top point. So, we have our base (4 meters) and our height (5 meters). Let’s plug these values into our formula: Area = 1/2 * 4 meters * 5 meters. First, we multiply 4 by 5, which gives us 20. Then, we take half of that (multiply by 1/2), which is 10. So, the area of one triangular face is 10 square meters. That’s great progress! We know how much fabric is needed for one triangle, but remember, there are four of them. We’re getting closer to that final answer, so let's keep pushing forward!

Calculating the Total Area of the Triangular Faces

We've figured out the area of one triangular face, which is 10 square meters. But Henri's lantern isn't just one triangle; it has four! All these triangles are identical, so calculating the total area is super straightforward. We just need to multiply the area of one triangle by the number of triangles. So, we're doing 10 square meters (the area of one triangle) * 4 (the number of triangles). This gives us a grand total of 40 square meters. That means Henri needs 40 square meters of fabric to cover all the triangular faces of his pyramid lantern. We’re really piecing this puzzle together now! We know the fabric needed for the base and the fabric needed for the triangles. What’s left? You guessed it – adding them up to get the total surface area. Let’s take this final step and nail the solution!

Finding the Total Surface Area

Okay, guys, we're in the home stretch! We've done all the hard work, and now it's time to bring it all together. We know that the square base needs 16 square meters of fabric, and the four triangular faces need a combined 40 square meters. To find the total amount of fabric Henri needs, we simply add these two amounts together. So, we’re doing 16 square meters (base) + 40 square meters (triangles). That gives us a grand total of 56 square meters. But wait a minute! Let's take a step back and look at our answer choices: A. $16 m^2$, B. $24 m^2$, C. $36 m^2$, and D. $40 m^2$. None of these match our calculated answer of 56 square meters! It seems there might be a little hiccup in the provided information or the answer choices. Based on our calculations, Henri needs 56 square meters of fabric. However, if we had to choose the closest answer from the options given, it would be D. $40 m^2$, but this is not the correct answer. It’s always good to double-check our work and the problem details to make sure everything lines up. Sometimes, a little discrepancy can remind us to be extra thorough in our problem-solving!

Final Answer

Alright, let’s wrap things up! After carefully calculating the area of the square base (16 square meters) and the total area of the four triangular faces (40 square meters), we found that Henri needs a total of 56 square meters of fabric to cover his square pyramid lantern. However, when we look at the provided answer choices (A. $16 m^2$, B. $24 m^2$, C. $36 m^2$, D. $40 m^2$), none of them match our calculated answer. This suggests there might be a slight issue with the options given or the initial problem details. If we were forced to choose the closest answer from the list, D. $40 m^2$ would be the nearest, but it's important to recognize that this isn't the accurate answer based on our calculations. It’s always a good practice to double-check and ensure all the information aligns. In this case, our detailed breakdown shows that 56 square meters is the correct amount of fabric needed. So, while the right answer isn't explicitly listed, we've confidently solved the problem using our geometry skills! Good job, guys!