Sometimes, you will need to find the surface area of a composite figure when a piece of a 3-dimensional figure has been removed.

Example

A cube with a cyndrical hole cut out of the middle of the cube is shown. The edge length of the cube is 12 meters, and the circle on the front face of the cube has a diameter of 4 feet. Find the surface area of the figure.

The steps needed to find the exposed surface area of the figure are:

  1. Determine the surface area of the Interactive popup. Assistance may be required. cube SA = 6s2
    = 6(12 feet)2
    = 864 ft2Close Pop Up     .
  2. Interactive popup. Assistance may be required. What shape The hole is a circle, and there are two of them (one on the front face, and one on the back face).Close Pop Up is the hole on the face of the cube, and how many holes are there?
  3. Determine theInteractive popup. Assistance may be required. combined area A = 2 x πr2
    ≈ 2 x 3.14(2 feet)2
    ≈ 25 ft2    Close Pop Up     of the holes.
  4. Determine the surface area of the cube without the holes. WhatInteractive popup. Assistance may be required. operation Subtraction Close Pop Up do you need to perform to remove the area of the holes?
  5. Determine theInteractive popup. Assistance may be required. resulting surface area SA = SAcube - Acircles
    ≈ 864 ft2 - 25 ft2
    ≈ 839 ft2     Close Pop Up     .
  6. WhatInteractive popup. Assistance may be required. part of the cylinder The lateral surface of the cylinder Close Pop Up is an exposed surface inside the cube?
  7. Find theInteractive popup. Assistance may be required. lateral area LA = 2πrh
    ≈ 2 (3.14)(2 feet)(12 feet)
    ≈ 151 ft2     Close Pop Up     of the cylinder.
  8. Interactive popup. Assistance may be required. Add SA = SAcube without holes + LAcylinder
    ≈ 839 ft2 + 151 ft2
    ≈ 990 ft2     Close Pop Up     the lateral area of the cylinder to the surface area of the cube without the holes.

It is worth noting that there are other ways you could have worked this problem. The important part is that you are sure that all surfaces that are exposed have been counted, and any removed surfaces—or removed parts of surfaces—have been subtracted.