Parallelepiped Surface Area: Step-by-Step Solution
Hey guys! Let's dive into a geometry problem where we need to find the total surface area of a right parallelepiped. The problem gives us some specific measurements, and we'll break it down step-by-step. So, grab your pencils and let's get started!
Understanding the Problem
So, the problem states: In a right parallelepiped, the sides of the base are 5 cm and 9 cm, forming an angle of 45°. The lateral edge is 8 cm. Find the total surface area of the prism. This sounds complicated, but it's totally manageable. We need to figure out what a right parallelepiped is and how to use the given info to calculate the total surface area.
A right parallelepiped is basically a prism where the base is a parallelogram, and all the side faces are rectangles because the lateral edges are perpendicular to the base. This makes our life a little easier since we don't have to deal with slanted sides when calculating areas. We're given the lengths of the two sides of the parallelogram base (5 cm and 9 cm), the angle between these sides (45°), and the height of the parallelepiped (8 cm), which is the length of the lateral edge.
Now, before we jump into calculations, let's think about what the total surface area actually consists of. It includes the areas of all the faces of the parallelepiped. Since it's a prism, we have two bases (each a parallelogram) and four lateral faces (rectangles). Our strategy will be to find the area of each of these faces and then add them all up to get the total surface area.
Calculating the Area of the Base
The base of our parallelepiped is a parallelogram with sides of 5 cm and 9 cm, and an angle of 45° between them. To find the area of a parallelogram, we use the formula:
Area = a * b * sin(θ)
Where:
aandbare the lengths of the sides,θis the angle between the sides.
In our case, a = 5 cm, b = 9 cm, and θ = 45°. Plugging these values into the formula, we get:
Area = 5 cm * 9 cm * sin(45°)
Remember that sin(45°) = √2 / 2, which is approximately 0.707. So,
Area = 5 cm * 9 cm * 0.707 ≈ 31.815 cm²
Since we have two bases, the total area of both bases is:
2 * 31.815 cm² ≈ 63.63 cm²
So, we've knocked out the bases! Now, let's move on to the lateral faces.
Calculating the Area of the Lateral Faces
The lateral faces of the right parallelepiped are rectangles. Since the parallelepiped is "right," these rectangles are formed by the sides of the parallelogram base and the lateral edge (height) of the parallelepiped. We have two pairs of rectangles: one pair with sides 5 cm and 8 cm, and another pair with sides 9 cm and 8 cm.
The area of a rectangle is simply:
Area = length * width
For the first pair of rectangles:
Area = 5 cm * 8 cm = 40 cm²
Since there are two such rectangles, their combined area is:
2 * 40 cm² = 80 cm²
For the second pair of rectangles:
Area = 9 cm * 8 cm = 72 cm²
Again, since there are two of these rectangles, their combined area is:
2 * 72 cm² = 144 cm²
Now, to find the total area of all the lateral faces, we add these two results together:
80 cm² + 144 cm² = 224 cm²
Alright, we've got the area of the lateral faces. Now, let's put it all together!
Calculating the Total Surface Area
To find the total surface area of the parallelepiped, we simply add the total area of the bases to the total area of the lateral faces:
Total Surface Area = Area of Bases + Area of Lateral Faces
We calculated the total area of the bases to be approximately 63.63 cm², and the total area of the lateral faces to be 224 cm². Therefore:
Total Surface Area = 63.63 cm² + 224 cm² ≈ 287.63 cm²
So, the total surface area of the right parallelepiped is approximately 287.63 square centimeters. That wasn't so bad, right?
Final Answer
The total surface area of the right parallelepiped is approximately 287.63 cm². Remember, it's all about breaking down the problem into smaller, manageable parts and understanding the properties of the geometric shapes involved.
Key Concepts Used
To solve this problem, we used a few key concepts from geometry:
- Area of a Parallelogram: The area of a parallelogram is given by the formula A = a * b * sin(θ), where a and b are the lengths of the sides, and θ is the angle between them. Understanding this formula is crucial for finding the area of the bases of the parallelepiped.
- Area of a Rectangle: The area of a rectangle is simply the product of its length and width, A = l * w. This was used to find the area of the lateral faces of the parallelepiped.
- Trigonometry (Sine Function): The sine function, specifically sin(45°), was used in calculating the area of the parallelogram base. Knowing common trigonometric values is very helpful in these types of problems.
- Properties of a Right Parallelepiped: Understanding that the lateral edges of a right parallelepiped are perpendicular to the base allows us to easily determine that the lateral faces are rectangles. This simplifies the area calculations.
- Total Surface Area: The total surface area of any prism or parallelepiped is the sum of the areas of all its faces. Breaking down the problem into finding the areas of individual faces and then summing them up is a fundamental strategy.
Understanding these concepts thoroughly will help you tackle similar geometry problems with confidence. Keep practicing, and you'll become a pro in no time!
Tips for Solving Similar Problems
When faced with similar geometry problems, here are some tips that can help you:
- Draw a Diagram: Always start by drawing a clear diagram of the figure described in the problem. Label all the given dimensions and angles. This visual representation can make it much easier to understand the problem and identify the relevant formulas.
- Break Down the Problem: Decompose the complex shape into simpler shapes for which you know the area formulas. In this case, we broke down the parallelepiped into parallelograms and rectangles.
- Identify Key Information: Carefully read the problem statement and identify all the given information. Determine what you need to find and what formulas or theorems might be relevant.
- Use Trigonometry: Be prepared to use trigonometric functions (sine, cosine, tangent) when dealing with angles. Remember common trigonometric values for angles like 30°, 45°, and 60°.
- Double-Check Your Calculations: After you've found the areas of individual faces, double-check your calculations to ensure accuracy. A small mistake can lead to a wrong final answer.
- Pay Attention to Units: Always include the correct units in your answer. In this case, the area is measured in square centimeters (cm²).
- Practice Regularly: The more you practice solving geometry problems, the more comfortable you'll become with the formulas and techniques involved. Look for practice problems in textbooks or online resources.
By following these tips and practicing regularly, you can improve your problem-solving skills and tackle even the most challenging geometry problems with confidence. Keep up the great work, and you'll be a geometry whiz in no time!
Conclusion
Calculating the total surface area of a right parallelepiped involves understanding its properties, breaking down the problem into smaller parts, and applying the appropriate area formulas. By finding the areas of the bases and lateral faces and then summing them up, we can determine the total surface area. With practice and a solid understanding of key geometric concepts, you can confidently solve similar problems. So keep learning, keep practicing, and have fun with geometry!