Finding The Total Surface Area Of A Triangular Prism

Hey everyone! Today, we're going to dive into a geometry problem involving a triangular prism. Specifically, we'll learn how to calculate the total surface area of a right triangular prism. The problem gives us some key information: the side edge is 7 cm, the base is a right triangle with a hypotenuse of 10 cm and one leg (cathetus) of 6 cm. Ready to get started, guys? Let's break down this problem step-by-step. This is the kind of stuff that can seem a little tricky at first, but once you get the hang of it, it's totally manageable. We'll go through everything you need to know, from understanding the basics to applying the right formulas and solving the problem. So, grab your pencils, paper, and let's get those math brains working!

Understanding the Basics: What is a Triangular Prism?

Alright, before we jump into the calculations, let's make sure we're all on the same page about what a triangular prism actually is. Imagine a shape with two identical triangles as its bases, connected by three rectangles. That's a triangular prism in a nutshell, right? Think of it like a tent or a Toblerone chocolate bar – those are great examples of triangular prisms. In our specific problem, we're dealing with a right triangular prism. This means the side edges (the rectangles connecting the triangles) are perpendicular to the triangular bases. This is super important because it simplifies our calculations, allowing us to find the surface area a bit more easily.

So, what about the surface area? The total surface area of a 3D shape is basically the total area of all its faces added together. For a triangular prism, this means we need to find the area of the two triangular bases and the three rectangular sides. Each of the rectangles will have one side length that is the same as the height of the prism (in our case, the side edge of 7 cm), and the other side length will be one of the sides of the triangle. Understanding this basic geometry is crucial before we start. This problem might sound complicated at first, but don't worry, we are going to simplify all steps to make the task clear.

Step 1: Find the Missing Side of the Triangle

Okay, let's get down to business! We know the hypotenuse (10 cm) and one leg (6 cm) of the right-angled triangle that forms the base of the prism. We need to find the length of the other leg. This is where the Pythagorean theorem comes in handy! The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). In our case, the hypotenuse (c) is 10 cm, and one leg (a) is 6 cm. We need to find the other leg (b).

Let's plug in the values and solve: 6² + b² = 10². This simplifies to 36 + b² = 100. Next, subtract 36 from both sides to get b² = 64. Taking the square root of both sides, we find that b = 8 cm. So, the missing side of the triangle is 8 cm. Great job, everyone! You just used the Pythagorean theorem to find a missing side. Now we have all the information about the triangle we need.

Step 2: Calculate the Area of the Triangular Base

Now that we know all the sides of the triangular base (6 cm, 8 cm, and 10 cm), we can calculate its area. The area of a triangle is given by the formula: (1/2) * base * height. In a right-angled triangle, the two legs can be considered the base and the height. So, in our case, the base is 6 cm, and the height is 8 cm (or vice versa, it doesn't matter). Therefore, the area of one triangular base is (1/2) * 6 cm * 8 cm = 24 cm². Remember, we have two such triangles in a prism!

Since the prism has two identical triangular bases, we need to consider the area of both. The combined area of the two bases will be 2 * 24 cm² = 48 cm². This gives us the area for the two bases, we need the area of the three rectangular sides to get the total area. You're doing great, keep it up! Next up, we will find the area of the rectangles, which will be the side surfaces of the prism.

Step 3: Calculate the Area of the Rectangular Sides

Alright, time to move on to the rectangles! The prism has three rectangular sides. The area of each rectangle is given by its length multiplied by its width. The length of each rectangle is the same as the side edge of the prism, which is given as 7 cm. The width of each rectangle is the length of the sides of the triangular base (6 cm, 8 cm, and 10 cm).

So, let's calculate the area of each rectangle:

• Rectangle 1: 6 cm * 7 cm = 42 cm²
• Rectangle 2: 8 cm * 7 cm = 56 cm²
• Rectangle 3: 10 cm * 7 cm = 70 cm².

Now we have calculated the area of the three rectangular sides. Now that we've found the area of each of the rectangles, we simply add them together to get the total area of all the rectangular sides: 42 cm² + 56 cm² + 70 cm² = 168 cm². We're almost there! It's just a matter of combining everything we've calculated so far. We are doing great!

Step 4: Calculate the Total Surface Area

We're in the home stretch, guys! Now we have all the pieces of the puzzle: the total area of the two triangular bases (48 cm²) and the total area of the three rectangular sides (168 cm²).

To find the total surface area of the prism, we simply add these two values together: 48 cm² + 168 cm² = 216 cm². And there you have it! The total surface area of the right triangular prism is 216 cm². We have successfully found the total surface area of the prism by breaking the task into multiple steps. Feel proud of the results, you did great!

Conclusion: Wrapping it Up

And that's a wrap! You've successfully calculated the total surface area of a right triangular prism. We started with the basic understanding of the shape, found the missing side using the Pythagorean theorem, calculated the area of the triangular bases, calculated the area of the rectangles, and finally added everything up to get the total surface area. Remember, the key is to break down the problem into smaller, more manageable steps. Don't be afraid to draw diagrams – they can be super helpful in visualizing the problem and understanding the relationships between the different parts.

Geometry can be a lot of fun, especially when you understand the steps involved. Keep practicing, and you'll become a pro in no time! If you have any questions, feel free to ask. Thanks for following along, and happy calculating! Keep up the great work, everyone. If you have similar tasks, feel free to try to repeat this procedure yourself, it will help you a lot!