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surface area of composite 3d figures quick checkQuestion
Surface area of composite 3-D figures quick check
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GPT-4o mini
To calculate the surface area of composite 3D figures, you typically follow these steps:
1. **Identify the Shapes**: Break the figure into its component shapes (e.g., cylinders, spheres, cones, prisms, etc.).
2. **Calculate Individual Areas**: Find the surface area for each individual shape using the appropriate formula:
- **Cube**: \( SA = 6a^2 \) (where \( a \) is the length of a side)
- **Rectangular Prism**: \( SA = 2lw + 2lh + 2wh \) (where \( l \), \( w \), and \( h \) are the length, width, and height)
- **Cylinder**: \( SA = 2\pi r^2 + 2\pi rh \) (where \( r \) is the radius and \( h \) is the height)
- **Sphere**: \( SA = 4\pi r^2 \) (where \( r \) is the radius)
- **Cone**: \( SA = \pi r^2 + \pi r l \) (where \( r \) is the radius and \( l \) is the slant height)
3. **Subtract Overlapping Areas**: If any surfaces are shared between the composite shapes (e.g., a cylinder sitting on top of a cone), subtract those overlapping areas from the total.
4. **Sum Areas**: Add the surface areas of all the individual shapes, considering any areas that need to be subtracted.
### Example:
**Find the surface area of a cylinder with a height of 10 cm and a radius of 3 cm that has a cone with a radius of 3 cm and a height of 4 cm on top.**
1. **Surface Area of the Cylinder**:
- Circle ends: \( 2\pi(3^2) = 18\pi \)
- Lateral area: \( 2\pi(3)(10) = 60\pi \)
- Total SA of cylinder: \( 18\pi + 60\pi = 78\pi \)
2. **Surface Area of the Cone**:
- Base (not included in total): \( \pi(3^2) = 9\pi \)
- Lateral area: Use slant height \( l \) given by \( l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \)
- Lateral SA: \( \pi(3)(5) = 15\pi \)
- Total SA of cone (excluding the base): \( 15\pi \)
3. **Composite Figure Total Surface Area**:
- Total = Cylinder SA + Cone Lateral SA
- Total = \( 78\pi + 15\pi = 93\pi \)
So, the surface area of the composite figure is \( 93\pi \) square centimeters.
Feel free to ask for further clarification or for more examples!
1. **Identify the Shapes**: Break the figure into its component shapes (e.g., cylinders, spheres, cones, prisms, etc.).
2. **Calculate Individual Areas**: Find the surface area for each individual shape using the appropriate formula:
- **Cube**: \( SA = 6a^2 \) (where \( a \) is the length of a side)
- **Rectangular Prism**: \( SA = 2lw + 2lh + 2wh \) (where \( l \), \( w \), and \( h \) are the length, width, and height)
- **Cylinder**: \( SA = 2\pi r^2 + 2\pi rh \) (where \( r \) is the radius and \( h \) is the height)
- **Sphere**: \( SA = 4\pi r^2 \) (where \( r \) is the radius)
- **Cone**: \( SA = \pi r^2 + \pi r l \) (where \( r \) is the radius and \( l \) is the slant height)
3. **Subtract Overlapping Areas**: If any surfaces are shared between the composite shapes (e.g., a cylinder sitting on top of a cone), subtract those overlapping areas from the total.
4. **Sum Areas**: Add the surface areas of all the individual shapes, considering any areas that need to be subtracted.
### Example:
**Find the surface area of a cylinder with a height of 10 cm and a radius of 3 cm that has a cone with a radius of 3 cm and a height of 4 cm on top.**
1. **Surface Area of the Cylinder**:
- Circle ends: \( 2\pi(3^2) = 18\pi \)
- Lateral area: \( 2\pi(3)(10) = 60\pi \)
- Total SA of cylinder: \( 18\pi + 60\pi = 78\pi \)
2. **Surface Area of the Cone**:
- Base (not included in total): \( \pi(3^2) = 9\pi \)
- Lateral area: Use slant height \( l \) given by \( l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \)
- Lateral SA: \( \pi(3)(5) = 15\pi \)
- Total SA of cone (excluding the base): \( 15\pi \)
3. **Composite Figure Total Surface Area**:
- Total = Cylinder SA + Cone Lateral SA
- Total = \( 78\pi + 15\pi = 93\pi \)
So, the surface area of the composite figure is \( 93\pi \) square centimeters.
Feel free to ask for further clarification or for more examples!
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