Question
Unit 6 Lesson 8 Surface Area of Composite 3D Figures
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GPT-4o mini
To find the surface area of composite 3D figures, you will typically follow a few steps. A composite 3D figure is made up of two or more simple 3D shapes, such as cubes, cylinders, cones, prisms, or spheres. Here’s a step-by-step guide to calculate the surface area:
### Steps to Calculate Surface Area of Composite 3D Figures
1. **Identify the Shapes**: Break down the composite figure into its individual shapes. For example, a figure may consist of a cylinder with a cone on top of it.
2. **Calculate the Surface Area of Each Shape**: Use the formulas for the surface area of each individual shape:
- **Cube**: \(SA = 6s^2\) (where \(s\) is the length of a side)
- **Rectangular Prism**: \(SA = 2lw + 2lh + 2wh\) (where \(l\) is length, \(w\) is width, and \(h\) is height)
- **Cylinder**: \(SA = 2\pi r^2 + 2\pi rh\) (where \(r\) is the radius and \(h\) is the height)
- **Cone**: \(SA = \pi r^2 + \pi r l\) (where \(r\) is the radius and \(l\) is the slant height)
- **Sphere**: \(SA = 4\pi r^2\)
3. **Identify Overlapping Areas**: When combining shapes, some parts may not contribute to the outer surface area (for example, the base of a cone sitting on top of a cylinder). Subtract the areas of these overlapping surfaces from the total surface area calculation.
4. **Sum the Surface Areas**: After calculating the surface areas of all individual shapes and adjusting for any overlaps, sum them up to get the total surface area of the composite figure.
### Example Problem
**Find the surface area of a composite figure made of a cylinder with a height of 10 cm and a radius of 3 cm, topped with a cone that has the same radius and a height of 4 cm.**
1. **Identify Shapes**: The figure is made of one cylinder and one cone.
2. **Calculate the Surface Area of Each Shape**:
- **Cylinder**:
\[
SA_{cylinder} = 2\pi r^2 + 2\pi rh = 2\pi(3^2) + 2\pi(3)(10) = 18\pi + 60\pi = 78\pi \, \text{cm}^2
\]
- **Cone** (do not include the base area as it sits on top of the cylinder):
\[
SA_{cone} = \pi r^2 + \pi r l
\]
First, find \(l\) (the slant height):
\[
l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Now, calculate the surface area:
\[
SA_{cone} = \pi(3^2) + \pi(3)(5) = 9\pi + 15\pi = 24\pi \, \text{cm}^2
\]
3. **Adjust for Overlap**:
- The base of the cone that overlaps with the top of the cylinder does not contribute to the external surface area:
\[
\text{Area of the base of the cone} = \pi r^2 = \pi(3^2) = 9\pi \, \text{cm}^2
\]
4. **Total Surface Area Calculation**:
\[
SA_{total} = SA_{cylinder} + SA_{cone} - \text{Area of base of cone}
\]
\[
SA_{total} = 78\pi + 24\pi - 9\pi = 93\pi \, \text{cm}^2
\]
### Conclusion
Thus, the total surface area of the composite figure is \(93\pi\) cm², which is approximately \(292.48\) cm² when evaluated numerically using \(\pi \approx 3.14\).
By following these steps and understanding how to handle overlapping areas, you can effectively calculate the surface area of any composite 3D figure.
### Steps to Calculate Surface Area of Composite 3D Figures
1. **Identify the Shapes**: Break down the composite figure into its individual shapes. For example, a figure may consist of a cylinder with a cone on top of it.
2. **Calculate the Surface Area of Each Shape**: Use the formulas for the surface area of each individual shape:
- **Cube**: \(SA = 6s^2\) (where \(s\) is the length of a side)
- **Rectangular Prism**: \(SA = 2lw + 2lh + 2wh\) (where \(l\) is length, \(w\) is width, and \(h\) is height)
- **Cylinder**: \(SA = 2\pi r^2 + 2\pi rh\) (where \(r\) is the radius and \(h\) is the height)
- **Cone**: \(SA = \pi r^2 + \pi r l\) (where \(r\) is the radius and \(l\) is the slant height)
- **Sphere**: \(SA = 4\pi r^2\)
3. **Identify Overlapping Areas**: When combining shapes, some parts may not contribute to the outer surface area (for example, the base of a cone sitting on top of a cylinder). Subtract the areas of these overlapping surfaces from the total surface area calculation.
4. **Sum the Surface Areas**: After calculating the surface areas of all individual shapes and adjusting for any overlaps, sum them up to get the total surface area of the composite figure.
### Example Problem
**Find the surface area of a composite figure made of a cylinder with a height of 10 cm and a radius of 3 cm, topped with a cone that has the same radius and a height of 4 cm.**
1. **Identify Shapes**: The figure is made of one cylinder and one cone.
2. **Calculate the Surface Area of Each Shape**:
- **Cylinder**:
\[
SA_{cylinder} = 2\pi r^2 + 2\pi rh = 2\pi(3^2) + 2\pi(3)(10) = 18\pi + 60\pi = 78\pi \, \text{cm}^2
\]
- **Cone** (do not include the base area as it sits on top of the cylinder):
\[
SA_{cone} = \pi r^2 + \pi r l
\]
First, find \(l\) (the slant height):
\[
l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Now, calculate the surface area:
\[
SA_{cone} = \pi(3^2) + \pi(3)(5) = 9\pi + 15\pi = 24\pi \, \text{cm}^2
\]
3. **Adjust for Overlap**:
- The base of the cone that overlaps with the top of the cylinder does not contribute to the external surface area:
\[
\text{Area of the base of the cone} = \pi r^2 = \pi(3^2) = 9\pi \, \text{cm}^2
\]
4. **Total Surface Area Calculation**:
\[
SA_{total} = SA_{cylinder} + SA_{cone} - \text{Area of base of cone}
\]
\[
SA_{total} = 78\pi + 24\pi - 9\pi = 93\pi \, \text{cm}^2
\]
### Conclusion
Thus, the total surface area of the composite figure is \(93\pi\) cm², which is approximately \(292.48\) cm² when evaluated numerically using \(\pi \approx 3.14\).
By following these steps and understanding how to handle overlapping areas, you can effectively calculate the surface area of any composite 3D figure.