Question
Math 7 B: Surface Area of Composite 3D figures Quick check
Answers
GPT-4o mini
To calculate the surface area of composite 3D figures, follow these general steps:
1. **Identify the Shapes**: Break down the composite figure into its basic geometric components (e.g., cylinders, cubes, spheres, etc.).
2. **Calculate the Surface Area for Each Shape**: Use the appropriate formulas for the surface area of each individual shape. Here are some common formulas:
- **Cube**: \(SA = 6s^2\) (where \(s\) is the length of one side)
- **Rectangular Prism**: \(SA = 2lw + 2lh + 2wh\) (where \(l\), \(w\), and \(h\) are length, width, and height, respectively)
- **Cylinder**: \(SA = 2\pi r(h + r)\) (where \(r\) is the radius and \(h\) is the height)
- **Sphere**: \(SA = 4\pi r^2\) (where \(r\) is the radius)
- **Cone**: \(SA = \pi r(r + l)\) (where \(r\) is the radius and \(l\) is the slant height)
3. **Adjust for Overlapping Areas**: If any parts of the figures overlap or are not exposed, you need to subtract the area of those overlapping parts from the total surface area.
4. **Add the Areas Together**: Sum the surface areas of all the individual shapes while ensuring that you've accounted for any overlaps.
### Example Problem
Imagine a composite figure made of a cylinder with a height of 10 cm and a radius of 3 cm, topped with a hemisphere of the same radius.
1. **Surface Area of the Cylinder**:
\[
SA_{cylinder} = 2\pi r(h + r) = 2\pi(3)(10 + 3) = 2\pi(3)(13) = 78\pi \text{ cm}^2
\]
2. **Surface Area of the Hemisphere**:
Since the flat circular base of the hemisphere sits on top of the cylinder, we only include the curved surface area:
\[
SA_{hemisphere} = 2\pi r^2 = 2\pi(3^2) = 18\pi \text{ cm}^2
\]
3. **Total Surface Area**:
We add the surface areas but subtract the area of the base of the hemisphere, which is \( \pi r^2 \):
\[
SA_{total} = SA_{cylinder} + SA_{hemisphere} - \text{Area of base of hemisphere}
\]
\[
= 78\pi + 18\pi - 9\pi = 87\pi \text{ cm}^2
\]
4. **Final Answer**:
The total surface area of the composite figure is \( 87\pi \text{ cm}^2 \) or approximately \( 273.66 \text{ cm}^2 \) (using \( \pi \approx 3.14 \)).
Feel free to provide an example of a composite figure or any specific scenarios you want help with, and I can assist you further!
1. **Identify the Shapes**: Break down the composite figure into its basic geometric components (e.g., cylinders, cubes, spheres, etc.).
2. **Calculate the Surface Area for Each Shape**: Use the appropriate formulas for the surface area of each individual shape. Here are some common formulas:
- **Cube**: \(SA = 6s^2\) (where \(s\) is the length of one side)
- **Rectangular Prism**: \(SA = 2lw + 2lh + 2wh\) (where \(l\), \(w\), and \(h\) are length, width, and height, respectively)
- **Cylinder**: \(SA = 2\pi r(h + r)\) (where \(r\) is the radius and \(h\) is the height)
- **Sphere**: \(SA = 4\pi r^2\) (where \(r\) is the radius)
- **Cone**: \(SA = \pi r(r + l)\) (where \(r\) is the radius and \(l\) is the slant height)
3. **Adjust for Overlapping Areas**: If any parts of the figures overlap or are not exposed, you need to subtract the area of those overlapping parts from the total surface area.
4. **Add the Areas Together**: Sum the surface areas of all the individual shapes while ensuring that you've accounted for any overlaps.
### Example Problem
Imagine a composite figure made of a cylinder with a height of 10 cm and a radius of 3 cm, topped with a hemisphere of the same radius.
1. **Surface Area of the Cylinder**:
\[
SA_{cylinder} = 2\pi r(h + r) = 2\pi(3)(10 + 3) = 2\pi(3)(13) = 78\pi \text{ cm}^2
\]
2. **Surface Area of the Hemisphere**:
Since the flat circular base of the hemisphere sits on top of the cylinder, we only include the curved surface area:
\[
SA_{hemisphere} = 2\pi r^2 = 2\pi(3^2) = 18\pi \text{ cm}^2
\]
3. **Total Surface Area**:
We add the surface areas but subtract the area of the base of the hemisphere, which is \( \pi r^2 \):
\[
SA_{total} = SA_{cylinder} + SA_{hemisphere} - \text{Area of base of hemisphere}
\]
\[
= 78\pi + 18\pi - 9\pi = 87\pi \text{ cm}^2
\]
4. **Final Answer**:
The total surface area of the composite figure is \( 87\pi \text{ cm}^2 \) or approximately \( 273.66 \text{ cm}^2 \) (using \( \pi \approx 3.14 \)).
Feel free to provide an example of a composite figure or any specific scenarios you want help with, and I can assist you further!