Prism Area & Volume: Step-by-Step Calculation Guide

Hey guys! Today, we're diving into the world of geometry to tackle a fun problem: figuring out the total area and volume of a hexagonal prism. Specifically, we're looking at a prism that's 50mm tall and has a regular hexagonal base with sides measuring 15mm. Sounds a bit complex? Don't worry, we'll break it down step by step so it's super easy to follow. Let's get started!

Understanding the Hexagonal Prism

Before we jump into calculations, let's quickly understand what we're dealing with. A hexagonal prism is a 3D shape with two hexagonal bases and six rectangular sides. Think of it like a stretched-out hexagon! To find its total area and volume, we need to consider the dimensions of its hexagonal base and its height. Our prism has a height of 50mm, and each side of the hexagonal base is 15mm. This information is crucial, so keep it in mind as we move forward.

Now, why is understanding this important? Well, the total area involves the area of the two hexagonal bases plus the area of the six rectangular sides. The volume, on the other hand, is the area of the hexagonal base multiplied by the prism's height. So, our first task is to figure out the area of that hexagon. Remember, we're dealing with a regular hexagon, which means all its sides and angles are equal. This makes our calculations a bit simpler. We'll explore the formula for the area of a regular hexagon in the next section, so stay tuned! We're going to make this as clear as possible, so you can confidently calculate the area and volume of any hexagonal prism you come across.

Calculating the Area of the Hexagonal Base

Alright, let's get to the heart of the problem: finding the area of the hexagonal base. Since it's a regular hexagon, we can use a nifty formula that makes our lives much easier. The formula is: Area = (3√3 / 2) * side², where “side” is the length of one side of the hexagon. In our case, the side length is 15mm. So, let's plug that into the formula and see what we get.

Area = (3√3 / 2) * (15mm)²

First, we calculate 15 squared (15 * 15), which equals 225. So now we have:

Area = (3√3 / 2) * 225

Next, we need to tackle the (3√3 / 2) part. √3 (the square root of 3) is approximately 1.732. So, 3√3 is about 3 * 1.732, which equals 5.196. Then, we divide that by 2: 5.196 / 2 = 2.598. Now we can substitute that back into our equation:

Area = 2.598 * 225

Finally, we multiply 2.598 by 225, which gives us approximately 584.55. Remember, this is the area in square millimeters (mm²), since our side length was in millimeters. So, the area of one hexagonal base is roughly 584.55 mm². But hold on, we're not done yet! We need this area to calculate both the total surface area and the volume of the prism, so keep this number handy. In the next section, we'll use this to find the total surface area, which includes the two hexagonal bases and the rectangular sides.

Determining the Lateral Area

Now that we've conquered the area of the hexagonal base, let's move on to finding the lateral area. The lateral area is simply the combined area of all the rectangular sides of the prism. Since our prism has a regular hexagonal base, it has six identical rectangular sides. To find the area of one rectangle, we multiply its length by its width. In this case, the length is the height of the prism (50mm), and the width is the side length of the hexagon (15mm).

So, the area of one rectangular side is:

Area of one rectangle = length * width = 50mm * 15mm = 750 mm²

Since there are six identical rectangles, we multiply the area of one rectangle by 6 to get the total lateral area:

Lateral Area = 6 * 750 mm² = 4500 mm²

Great! We've found the lateral area, which is a significant piece of the puzzle. This tells us the total area covered by the sides of the prism. Now, remember that total surface area we talked about earlier? To find that, we need to add the lateral area to the area of the two hexagonal bases. We already calculated the area of one base, so we're well on our way. In the next section, we'll put it all together to calculate the total surface area and then move on to the volume. Keep up the great work; we're almost there!

Calculating the Total Surface Area

Okay, guys, let's calculate the total surface area of our hexagonal prism. We've already done the groundwork by finding the area of one hexagonal base and the lateral area. Remember, the total surface area is the sum of the areas of all the faces of the prism. That means we need to add the areas of the two hexagonal bases to the lateral area.

We calculated the area of one hexagonal base to be approximately 584.55 mm². Since we have two bases, the total area of the bases is:

Total Base Area = 2 * 584.55 mm² = 1169.1 mm²

We also found the lateral area to be 4500 mm². Now, we simply add the total base area and the lateral area to get the total surface area:

Total Surface Area = Total Base Area + Lateral Area

Total Surface Area = 1169.1 mm² + 4500 mm² = 5669.1 mm²

So, the total surface area of our hexagonal prism is approximately 5669.1 square millimeters. That's a pretty big number, but it makes sense when you consider all the faces of the prism we're adding up. We're almost there! We've tackled the surface area, and now we just have one more calculation to go: the volume. In the next section, we'll use the base area we calculated earlier and the height of the prism to find the volume. Get ready to wrap this up!

Calculating the Volume of the Prism

Alright, let's wrap things up by calculating the volume of our hexagonal prism! The volume of a prism tells us how much space it occupies. The formula for the volume of any prism is quite straightforward: Volume = Base Area * Height. We already have both of these values for our prism. We calculated the area of the hexagonal base to be approximately 584.55 mm², and we know the height of the prism is 50mm.

So, let's plug these values into the formula:

Volume = 584.55 mm² * 50 mm

Volume = 29227.5 mm³

Therefore, the volume of our hexagonal prism is approximately 29227.5 cubic millimeters (mm³). Notice that the unit is cubic millimeters because we're measuring volume, which is a three-dimensional quantity. And that's it! We've successfully calculated the volume of our hexagonal prism. We've come a long way, from understanding the prism's shape to calculating the area of its base, lateral area, total surface area, and finally, the volume.

Conclusion: Mastering Prism Calculations

Woohoo! We've made it to the end, guys! Calculating the total area and volume of a hexagonal prism might seem daunting at first, but as we've seen, it's totally manageable when you break it down into smaller steps. We started by understanding the shape, then calculated the area of the hexagonal base, moved on to the lateral area, combined those to find the total surface area, and finally, calculated the volume. Each step built upon the previous one, making the whole process much clearer.

The key takeaways here are the formulas: Area of a regular hexagon = (3√3 / 2) * side², Lateral Area = perimeter of base * height, Total Surface Area = 2 * Base Area + Lateral Area, and Volume = Base Area * Height. Keep these in your toolbox, and you'll be ready to tackle any prism problem that comes your way.

Remember, practice makes perfect. Try working through similar problems with different dimensions to solidify your understanding. Geometry can be a lot of fun once you get the hang of it, and understanding these calculations opens the door to more advanced concepts. So, keep exploring, keep learning, and most importantly, keep having fun with math! You've got this!