Ask a New Question
Menu

Math 7 B: Surface Area of Composite 3D figures Quick check

1 answer

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!