Calculate Total Area And Volume Of Hexagonal Prism

by Omar Yusuf 51 views

Hey guys! Today, we're diving into the fascinating world of geometry to tackle a problem that involves calculating the total area and volume of a prism with a regular hexagonal base. Don't worry, it sounds more complicated than it actually is. We'll break it down step by step, so you'll be a pro in no time! Let's get started and unlock the secrets of this geometric puzzle.

Understanding the Hexagonal Prism

Before we jump into calculations, let's make sure we all understand what a hexagonal prism actually is. Imagine a hexagon, that six-sided shape we all know and love. Now, picture taking that hexagon and stacking identical hexagons on top of each other, creating a three-dimensional shape. That, my friends, is a hexagonal prism! It's a prism because it has two identical bases (the hexagons) connected by rectangular faces. Think of it like a honeycomb, but with perfectly flat faces. To really grasp this, consider that in our specific case, this regular hexagon boasts sides measuring 5 cm, a nifty apothem of 3.64 cm, and the entire prism stands tall at a height of 4.2 cm. These measurements are our golden tickets to unlocking the area and volume, so keep them in mind as we proceed. Having a clear visual of what we're working with is crucial, so take a moment to picture this in your mind.

Calculating the Area of the Hexagonal Base

The first step in finding the total area and volume of our prism is to figure out the area of its hexagonal base. Now, hexagons might seem intimidating, but there's a neat trick to calculating their area. Remember that a regular hexagon is made up of six identical equilateral triangles? We can use this to our advantage! To find the area of one of these triangles, we need its base and height. The base is simply the side length of the hexagon, which we know is 5 cm. The height of the triangle is the apothem of the hexagon, which is given as 3.64 cm. The area of a triangle is (1/2) * base * height, so for one of our triangles, it's (1/2) * 5 cm * 3.64 cm = 9.1 cm². But hold on, we have six of these triangles! So, the area of the entire hexagonal base is 6 * 9.1 cm² = 54.6 cm². This is a crucial piece of the puzzle, so make sure you've got this calculation down. Remember, breaking down complex shapes into simpler ones is a key strategy in geometry.

Calculating the Lateral Area of the Prism

Now that we've conquered the base area, let's move on to the lateral area. The lateral area is simply the area of all the rectangular faces that connect the two hexagonal bases. Imagine unfolding the prism like a cardboard box – the lateral area is what you'd see laid out flat, excluding the top and bottom. Each of these rectangular faces has a length equal to the side of the hexagon (5 cm) and a width equal to the height of the prism (4.2 cm). So, the area of one rectangular face is 5 cm * 4.2 cm = 21 cm². How many rectangular faces do we have? Since a hexagon has six sides, we have six rectangular faces. Therefore, the total lateral area is 6 * 21 cm² = 126 cm². This calculation is quite straightforward, but it's important to understand what the lateral area represents. It's the surface area of the prism excluding the bases, and it's a key component of the total surface area.

Calculating the Total Area of the Prism

We're almost there! We've calculated the area of the hexagonal base and the lateral area. To find the total area of the prism, we just need to put these pieces together. The total area is the sum of the areas of the two bases and the lateral area. We know the area of one hexagonal base is 54.6 cm², so the area of both bases is 2 * 54.6 cm² = 109.2 cm². We also know the lateral area is 126 cm². Therefore, the total area of the prism is 109.2 cm² + 126 cm² = 235.2 cm². Boom! We've done it! This is the total surface area of our hexagonal prism, the amount of material you'd need to cover the entire shape. Remember, it's all about breaking down the problem into smaller, manageable steps. We found the base area, the lateral area, and then simply added them together. Easy peasy!

Calculating the Volume of the Prism

Alright, guys, let's switch gears and tackle the volume of the hexagonal prism. Volume is the amount of space a three-dimensional object occupies. Think of it as the amount of water you could pour into the prism to fill it completely. The formula for the volume of any prism is super simple: Volume = Base Area * Height. We already know the area of the hexagonal base is 54.6 cm², and we know the height of the prism is 4.2 cm. So, the volume is simply 54.6 cm² * 4.2 cm = 229.32 cm³. That's it! We've calculated the volume. Notice that the units are cubic centimeters (cm³) because we're measuring a three-dimensional space. This calculation highlights a fundamental principle: volume is all about the base area extended through the height of the object. In essence, we're finding out how many of those 54.6 cm² hexagonal layers we need to stack up to reach a height of 4.2 cm. It's like building a tower with hexagonal tiles!

Putting it All Together: The Grand Finale

So, let's recap what we've accomplished. We started with a hexagonal prism with a base side of 5 cm, an apothem of 3.64 cm, and a height of 4.2 cm. We then broke down the problem into smaller parts: calculating the area of the hexagonal base, finding the lateral area, determining the total surface area, and finally, calculating the volume. We discovered that the total area of the prism is 235.2 cm², and the volume is 229.32 cm³. These two values give us a complete picture of the prism's size and the space it occupies. Remember, guys, geometry might seem intimidating at first, but by breaking down problems into smaller steps and understanding the underlying concepts, you can conquer any geometric challenge. Keep practicing, and you'll be a geometry whiz in no time!

Here’s a quick recap of the key concepts and keywords we covered in this article:

  • Hexagonal Prism: A prism with two hexagonal bases connected by rectangular faces.
  • Area of a Hexagon: Can be calculated by dividing the hexagon into six equilateral triangles and summing their areas.
  • Apothem: The distance from the center of a regular polygon to the midpoint of a side.
  • Lateral Area: The sum of the areas of the rectangular faces of a prism.
  • Total Area: The sum of the areas of all the faces of a prism (two bases plus the lateral area).
  • Volume: The amount of space a three-dimensional object occupies. For a prism, Volume = Base Area * Height.

Calculate Total Area and Volume of Hexagonal Prism Example