Calculating Surface Area And Volume Of Various 3D Shapes

Oct 9, 2025 by ADMIN 57 views

Hey guys! Let's dive into the world of 3D shapes and figure out how to calculate their surface area and volume. We'll be tackling some problems from Gambar 2.81, breaking down each shape, and making sure we understand the concepts. Get ready to flex those math muscles! Remember, understanding these concepts is super important in math, and it can also be a lot of fun. Let's get started!

Understanding Surface Area and Volume

Before we jump into the problems, let's quickly recap what surface area and volume actually mean. Surface area is the total area that covers the surface of a 3D object. Imagine you're wrapping a present – the surface area is the amount of wrapping paper you'd need. It's measured in square units (like cm² or m²). On the other hand, volume is the amount of space that a 3D object occupies. Think of it as the amount of water a container can hold. Volume is measured in cubic units (like cm³ or m³).

So, why is this important? Well, knowing the surface area is useful for things like calculating the amount of paint needed to cover a wall or how much material is needed to build a box. Calculating volume is important when you are dealing with the capacity of something, such as the amount of water in a pool or the amount of sand in a sandbox. Basically, knowing both surface area and volume gives us a better understanding of the properties of 3D objects and how they relate to the real world. Got it? Awesome! Now, let's solve the given problem.

We will utilize several key formulas throughout our calculations. For instance, to calculate the surface area of a rectangular prism, we use the formula: 2lw + 2lh + 2wh, where 'l' is length, 'w' is width, and 'h' is height. For the volume, we use lwh. Understanding and correctly applying these formulas is key to solving these problems. For other shapes, like cylinders and spheres, we would, of course, use different formulas. The important thing is to know which formula to use for the type of shape you are dealing with. Remember, always pay attention to the units! This is one of the most common mistakes, and it can make a huge difference in your answers. Always make sure to include the units with the answers.

Calculating Surface Area and Volume: Step-by-Step

Now, let's tackle the shapes presented in Gambar 2.81. We'll go through each one, breaking down the steps and making sure we understand the calculations. Remember, practice makes perfect, so don't be afraid to try these problems on your own first!

a. Rectangular Prism

This shape is a rectangular prism. We are given the following measurements: length = 10 m, width = 10 m, and height = 10 m. This is a cube (a special type of rectangular prism where all sides are equal), so it's super easy to calculate.

Surface Area: The formula for the surface area of a rectangular prism (or cube) is 6 * side². In this case, it is 6 * (10 m)² = 6 * 100 m² = 600 m². So, the surface area is 600 m².
Volume: The formula for the volume of a rectangular prism is length * width * height. In this case, it’s 10 m * 10 m * 10 m = 1000 m³. So, the volume is 1000 m³.

b. Cylinder

This shape appears to be a cylinder. The given information is: radius = 8 cm and height = 10 cm.

Surface Area: The surface area of a cylinder is calculated using the formula: 2πr² + 2πrh, where 'r' is the radius and 'h' is the height. Substituting the values, we get: 2π(8 cm)² + 2π(8 cm)(10 cm) = 2π(64 cm²) + 2π(80 cm²) ≈ 402.12 cm² + 502.65 cm² ≈ 904.77 cm². So, the surface area is approximately 904.77 cm².
Volume: The volume of a cylinder is calculated using the formula: πr²h. Substituting the values, we get: π(8 cm)²(10 cm) = π(64 cm²)(10 cm) ≈ 2009.6 cm³. So, the volume is approximately 2009.6 cm³.

c. Rectangular Prism

This is another rectangular prism. Measurements: length = 10 m, width = 10 m, and height = 10 m. This is the same as (a), a cube!

Surface Area: As we calculated before, the surface area is 6 * (10 m)² = 600 m².
Volume: The volume is 10 m * 10 m * 10 m = 1000 m³.

d. Triangular Prism

This is a triangular prism. Measurements: base of triangle = 6 cm, height of triangle = 4 cm, and length of prism = 20 cm.

Surface Area: The surface area of a triangular prism is calculated using the formula: (base * height) + (perimeter of triangle * length of prism). First, let's calculate the area of the two triangular faces: 2 * (0.5 * 6 cm * 4 cm) = 24 cm². Next, we must figure out the perimeter of the triangle, which has sides 6, 5, and 5. This makes the perimeter 6 cm + 5 cm + 5 cm = 16 cm. The rectangular faces’ area is the perimeter multiplied by the length, 16 cm * 20 cm = 320 cm². Adding these areas together, 24 cm² + 320 cm² = 344 cm². So, the surface area is 344 cm².
Volume: The volume of a triangular prism is calculated using the formula: (0.5 * base * height) * length. Substituting the values, we get: (0.5 * 6 cm * 4 cm) * 20 cm = 240 cm³. So, the volume is 240 cm³.

e. Rectangular Prism

This is another rectangular prism. Measurements: length = 60 cm, width = 20 cm, and height = 9.6 m. Let’s keep our units consistent; we will convert the height to cm, which is 960 cm.

Surface Area: The formula is 2lw + 2lh + 2wh, which gives us (2 * 60cm * 20cm) + (2 * 60cm * 960cm) + (2 * 20cm * 960cm) = 2400 cm² + 115200 cm² + 38400 cm² = 156000 cm². So, the surface area is 156000 cm².
Volume: The formula is lwh, which gives us 60 cm * 20 cm * 960 cm = 1152000 cm³. So, the volume is 1152000 cm³.

f. Missing shape, a cylinder

This appears to be a cylinder. Measurements: radius = 60 cm and height = 9.6 m, which we can convert to 960 cm.

Surface Area: The surface area of a cylinder is calculated using the formula: 2πr² + 2πrh, where 'r' is the radius and 'h' is the height. Substituting the values, we get: 2π(60 cm)² + 2π(60 cm)(960 cm) = 2π(3600 cm²) + 2π(57600 cm²) ≈ 22619.47 cm² + 361911.45 cm² ≈ 384530.92 cm². So, the surface area is approximately 384530.92 cm².
Volume: The volume of a cylinder is calculated using the formula: πr²h. Substituting the values, we get: π(60 cm)²(960 cm) = π(3600 cm²)(960 cm) ≈ 10857336.9 cm³. So, the volume is approximately 10857336.9 cm³.

Important Notes and Tips

Units: Always include the correct units (cm, m, cm², m², cm³, m³) in your answers. Not doing so can lead to mistakes. Make sure everything is in the same units before calculating. I cannot stress this enough!
Formulas: Memorize the basic formulas for common shapes or have them readily available. Understanding when to use each formula is just as important as knowing them.
Step-by-Step: Break down complex problems into smaller steps. This makes it easier to identify mistakes and understand the process. Visualizing the shape can also help.
Practice: The more you practice, the better you'll become! Try solving similar problems on your own. You can find many practice problems online. Don't be afraid to ask for help from a teacher or tutor if you are struggling. Math can be difficult, but persistence pays off!
Real-world applications: Think about how these concepts apply in real-world situations. This can make learning more engaging and help you understand why these calculations are useful.

Conclusion

And that's it, guys! We've successfully calculated the surface area and volume of the shapes presented in Gambar 2.81. Remember to practice, stay focused, and don't be afraid to ask for help. Keep up the great work, and you'll master these concepts in no time. Keep on learning and keep exploring the world of math; it's more useful than you might think. Thanks for joining me today!