Prism Calculations: Base, Area, And Perimeter Explained

Hey guys! Today, we're diving deep into the world of prisms, specifically a regular triangular prism. We've got a cool problem to solve, and by the end of this, you'll be a prism-calculating pro. Let's break it down step by step, making sure everyone understands the ins and outs. We'll be tackling the perimeter and area of the base, the lateral face, and even the perimeter of a special triangle within our prism. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we're all on the same page. We're dealing with a regular triangular prism, which is a prism with triangular bases that are equilateral triangles, and the lateral faces are rectangles. Think of it like a perfectly symmetrical Toblerone box – that's the vibe we're going for. The problem gives us two crucial pieces of information:

• The base edge is 6 cm. This means each side of our equilateral triangle base is 6 cm long.
• The lateral edge is 8 cm. This is the height of our prism, or the length of the rectangular sides.

With these dimensions in hand, we can calculate various properties of the prism. We'll be tackling the perimeter and area of the base, the perimeter and area of a lateral face, and finally, the perimeter of triangle ABC', which adds a little twist to the problem. Remember, guys, understanding the shape and its dimensions is half the battle!

a) Perimeter of the Base

Let's start with something simple: the perimeter of the base. Since our base is an equilateral triangle, all three sides are equal. We know each side is 6 cm, so finding the perimeter is a breeze. The perimeter of any shape is just the sum of the lengths of all its sides. In this case, we have three sides, each 6 cm long.

So, the calculation is straightforward:

Perimeter of base = 6 cm + 6 cm + 6 cm = 18 cm

Easy peasy, right? The perimeter of the base of our triangular prism is 18 cm. This is a fundamental measurement, and it's going to help us with other calculations down the line. Remember this, guys: the perimeter is simply the distance around the outside of a shape. For an equilateral triangle, it's three times the length of one side.

b) Area of the Base

Now, let's crank things up a notch and calculate the area of the base. This is where things get a little more interesting, but don't worry, we'll break it down. We're still dealing with an equilateral triangle, but this time we need to figure out the space it occupies. The formula for the area of an equilateral triangle is:

Area = (side² * √3) / 4

Where "side" is the length of one side of the triangle. We already know the side length is 6 cm, so we can plug that into our formula:

Area = (6² * √3) / 4

Let's simplify this step by step. First, 6² is 36, so we have:

Area = (36 * √3) / 4

Next, we can divide 36 by 4, which gives us 9:

Area = 9√3 cm²

So, the area of the base is 9√3 square centimeters. If you need a decimal approximation, √3 is about 1.732, so the area is approximately 9 * 1.732 = 15.588 cm². But leaving it as 9√3 cm² is perfectly accurate and often preferred in mathematical contexts. Guys, remember this formula; it's your best friend when dealing with equilateral triangles!

c) Perimeter of a Lateral Face

Okay, time to shift our focus from the base to the lateral faces of the prism. These are the rectangular sides that connect the two triangular bases. To find the perimeter of a lateral face, we need to consider its dimensions. We know the lateral edge (the height of the prism) is 8 cm, and the base edge (which is also the width of the rectangle) is 6 cm.

A rectangle has two pairs of equal sides, so the perimeter is simply twice the sum of the length and width:

Perimeter = 2 * (length + width)

In our case, the length is 8 cm and the width is 6 cm. Plugging those values in, we get:

Perimeter = 2 * (8 cm + 6 cm) = 2 * 14 cm = 28 cm

So, the perimeter of a lateral face is 28 cm. This one was pretty straightforward, right? Just remember the basic formula for the perimeter of a rectangle, and you're golden. Guys, this is a classic application of perimeter calculations, so make sure you've got it down!

d) Area of a Lateral Face

Next up, we're calculating the area of a lateral face. Since we've already established that the lateral faces are rectangles, this should be a piece of cake. The area of a rectangle is simply its length multiplied by its width:

Area = length * width

Again, we know the length is 8 cm and the width is 6 cm. So, the calculation is:

Area = 8 cm * 6 cm = 48 cm²

Therefore, the area of a lateral face is 48 square centimeters. See how understanding the basic shapes and their formulas makes these calculations so much easier? Guys, keep these fundamental formulas in your back pocket; they'll serve you well in all sorts of geometry problems.

e) Perimeter of Triangle ABC'

Alright, guys, this is where things get a little more interesting! We need to find the perimeter of triangle ABC'. This isn't just a face of the prism; it's a triangle formed by connecting two vertices of the base (A and B) with a vertex of the opposite base (C'). This means we need to figure out the lengths of all three sides of this triangle: AB, BC', and AC'.

We already know one side: AB is simply the base edge, which is 6 cm. Now, we need to find the lengths of BC' and AC'. Notice that BC' and AC' are actually the diagonals of the rectangular lateral faces. Since the lateral faces are congruent (identical), BC' and AC' will have the same length.

To find the length of BC' (or AC'), we can use the Pythagorean theorem. Remember, the Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our case, BC' is the hypotenuse of a right-angled triangle formed by the lateral edge (8 cm) and the base edge (6 cm).

So, we have:

BC'² = 8² + 6² = 64 + 36 = 100

Taking the square root of both sides, we get:

BC' = √100 = 10 cm

Since AC' is the same as BC', AC' is also 10 cm. Now we have all three sides of triangle ABC': AB = 6 cm, BC' = 10 cm, and AC' = 10 cm.

The perimeter of triangle ABC' is the sum of these sides:

Perimeter = 6 cm + 10 cm + 10 cm = 26 cm

So, the perimeter of triangle ABC' is 26 cm. This one required a little more thought and the application of the Pythagorean theorem, but we nailed it! Guys, this is a great example of how different geometric concepts can come together in a single problem.

Conclusion

And there you have it! We've successfully calculated the perimeter of the base (18 cm), the area of the base (9√3 cm²), the perimeter of a lateral face (28 cm), the area of a lateral face (48 cm²), and the perimeter of triangle ABC' (26 cm). We covered a lot of ground, from basic perimeter and area calculations to applying the Pythagorean theorem. Remember, guys, the key is to break down complex problems into smaller, manageable steps. Understanding the shapes and their properties is crucial, and practice makes perfect. Keep those formulas handy, and you'll be solving prism problems like a pro in no time!