To find the surface area of composite 3D figures, you'll typically need to break down the figure into its individual components, find the surface area of each component, and then combine those areas while accounting for any overlapping areas that should not be double-counted.
Here's a general approach you can follow:
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Identify the Components: Look at the composite figure and determine what shapes make it up (e.g., cylinders, cubes, spheres, cones).
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Calculate Individual Surface Areas:
- Use the appropriate formulas for each shape:
- Cube: Surface Area = 6s², where s is the length of a side.
- Rectangular Prism: Surface Area = 2lw + 2lh + 2wh, where l, w, and h are the length, width, and height respectively.
- Cylinder: Surface Area = 2πr² + 2πrh, where r is the radius and h is the height (includes both top and bottom).
- Sphere: Surface Area = 4πr², where r is the radius.
- Cone: Surface Area = πr² + πrl, where r is the radius of the base and l is the slant height.
- Use the appropriate formulas for each shape:
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Account for Overlapping Areas: If there are parts of the components that overlap (like a cylinder sitting on top of a base), you need to subtract the area of the overlapping section from the total.
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Sum the Areas: Add all the individual surface areas together, ensuring to include any modifications for overlaps.
Here is a quick example:
Example: A cylinder of radius 3 cm and height 5 cm is placed on top of a cube with side 4 cm. What is the total surface area?
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Calculate the surface area of the cube: \[ SA_{\text{cube}} = 6s^2 = 6(4)^2 = 96 , \text{cm}^2 \]
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Calculate the surface area of the cylinder: \[ SA_{\text{cylinder}} = 2\pi r^2 + 2\pi rh = 2\pi(3^2) + 2\pi(3)(5) = 18\pi + 30\pi = 48\pi , \text{cm}^2 \approx 150.8 , \text{cm}^2 \]
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Subtract the area of the top face of the cube that the cylinder covers: \[ \text{Area of the top of the cube} = s^2 = 4^2 = 16 , \text{cm}^2 \]
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So, the total surface area would be: \[ SA_{\text{total}} = SA_{\text{cube}} + SA_{\text{cylinder}} - \text{Area of overlap} \] \[ SA_{\text{total}} = 96 + 48\pi - 16 \approx 96 + 150.8 - 16 = 230.8 , \text{cm}^2 \]
Make sure to adjust your approach based on the specific figure you are working with!