Right Prism Calculations: Finding Diagonals And Surface Areas

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Hey guys! Let's dive into a geometry problem involving a right prism. We'll be working with a prism labeled ABCDA'B'C'D', where the base is a square, and we've got some angles to play with. Our goal is to calculate the length of a diagonal on one of the faces and then find the total surface area of the lateral faces. Sounds like fun, right?

Understanding the Problem

First off, let's break down what we're dealing with. We have a right prism, which means the sides are perpendicular to the bases. This is super important because it gives us right angles to work with, which is always a win in geometry. Our base is a square, specifically with a side length of 6 cm. A square base tells us all the sides are equal, and all the angles are right angles (90 degrees). And we know that the angle ∠C'BC is 60 degrees. This gives us a relationship between the side of the base and the height of the prism. Now let's look at what the question is asking us to do.

We need to find two things:

  • a) The length of the diagonal BC' of the face BCC'B'. This is where a little bit of 3D visualization comes in handy. Imagine the prism standing upright. The face BCC'B' is one of the rectangular sides. The line BC' cuts diagonally across this rectangle. This is what we need to measure. We know it creates a right triangle where we can use the Pythagorean theorem to work out the length.
  • b) The sum of the areas of the lateral faces of the prism. The lateral faces are the sides of the prism, excluding the top and bottom. Since our base is a square, all these lateral faces are rectangles. We'll need to calculate the area of one of these rectangles and then multiply by the number of rectangles to get the total area. In this case, we have 4 rectangles.

This problem gives us a great opportunity to flex our geometric muscles! It's all about understanding shapes, angles, and how to apply formulas. Let's get started and get these calculations done! I promise it's easier than it sounds, and you'll feel awesome when you're done.

Calculating the Length of the Diagonal BC'

Alright, let's tackle the first part of the problem: finding the length of the diagonal BC'. The key is to recognize that we have a right triangle formed by the sides BC, CC', and the diagonal BC'. Think of the face BCC'B' as a rectangle. The diagonal BC' splits that rectangle into two right triangles, each with BC and CC' as sides. We know the length of BC is equal to the side of the square base, so BC = 6 cm. We also know that ∠C'BC = 60°, and this is really useful information. We need to find the height, CC'.

We can start by finding the height of the prism, which is the length of CC'. Notice that triangle BCC' is a right triangle. We have an angle (60 degrees) and the length of one side (BC = 6cm), so we can use trigonometric ratios. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, tan(∠C'BC) = CC'/BC. We can rewrite this equation as CC' = BC * tan(60°). We know BC = 6 cm, and tan(60°) = √3. Thus, CC' = 6 * √3 cm. Now we have the height of the prism.

Now that we know the height CC', we can use the Pythagorean theorem to find BC'. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In triangle BCC', BC' is the hypotenuse. So, BC'² = BC² + CC'². We know BC = 6 cm and CC' = 6√3 cm. Plugging these values in gives us BC'² = 6² + (6√3)². This simplifies to BC'² = 36 + 108, which means BC'² = 144. Taking the square root of both sides, we get BC' = 12 cm. So the length of the diagonal BC' is 12 cm. Nice work, team!

This part of the problem is all about breaking down the 3D shape into manageable 2D pieces and applying the right formulas. It’s a great example of how geometry combines visualization with the right tools.

Calculating the Sum of the Lateral Faces Areas

Now, let's move on to the second part of the problem: finding the sum of the areas of the lateral faces. Remember, the lateral faces are the sides of the prism, and they're all rectangles in this case. The base of each rectangle is a side of the square base (6 cm), and the height of each rectangle is the height of the prism (CC' = 6√3 cm), which we calculated earlier. To find the area of one lateral face, we multiply the base by the height: Area = base * height = 6 cm * 6√3 cm = 36√3 cm².

Since our prism has a square base, it has four lateral faces. To find the total lateral surface area, we multiply the area of one face by four: Total Lateral Area = 4 * 36√3 cm² = 144√3 cm². Therefore, the sum of the areas of the lateral faces of the prism is 144√3 cm². And just like that, we've solved the second part of the problem.

This section highlights the importance of knowing the properties of the prism's shapes and knowing which measurements to use. Always be sure to correctly identify and apply the correct formulas. It is all about understanding the prism's structure and how the different parts relate to each other.

Conclusion

So, there you have it, guys! We've successfully calculated the length of the diagonal BC' (12 cm) and the sum of the areas of the lateral faces (144√3 cm²). We started with a geometric problem involving a right prism and broke it down step by step, applying our knowledge of geometry and basic trigonometry. We used the Pythagorean theorem to find the length of the diagonal and then used the dimensions of the faces to calculate the total area. Remember, geometry is all about understanding shapes and applying the right formulas to find missing information. Keep practicing, and you'll get better and better! You'll be acing these problems in no time!

Key Takeaways:

  • Right Prisms: Know the properties of right prisms, especially that the lateral faces are rectangles and the sides are perpendicular to the bases.
  • Pythagorean Theorem: Use the Pythagorean theorem to find missing sides in right triangles. This is a fundamental skill in geometry.
  • Trigonometric Ratios: Understand how to use sine, cosine, and tangent to find the sides and angles in right triangles. Remember the relationship: tan(angle) = opposite/adjacent.
  • Area Formulas: Memorize and apply the formulas for the area of rectangles and other shapes. Area = base * height.
  • Visualization: Practice visualizing 3D shapes to help you understand the relationships between their different parts.

Keep up the great work, and keep exploring the fascinating world of geometry! You’ve got this!