Surface area of composite 3-D figures quick check

Answer 1

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!