The GRE tests two particular three-dimensional shapes formed from polygons: the rectangular solid and the cube. Note that a cube is just a special type of rectangular solid:
The surface area of a three-dimensional shape is the amount of space on the surface of that particular object. For example, the amount of paint that it would take to fully cover a rectangular box could be determined by finding the surface area of that box. As with simple area, surface area is measured in square units such as in2 (square inches) or ft2 (square feet).
| Surface Area = the SUM of the areas of ALL of the faces |
Both a rectangular solid and a cube have six faces.
To determine the surface area of a rectangular solid, you must find the area of each face. Notice, however, that in a rectangular solid, the front and back faces have the same area, the top and bottom faces have the same area, and the two side faces have the same area. In the rectangular solid, the area of the front face is equal to 12 × 4 = 48. Thus, the back face also has an area of 48. The area of the bottom face is equal to 12 × 3 = 36. Thus, the top face also has an area of 36. Finally, each side face has an area of 3 × 4 = 12. Therefore, the surface area, or the sum of the areas of all six faces, is: 48(2) + 36(2) + 12(2) = 192.
To determine the surface area of a cube, you only need the length of one side. You can see from the preceding cube that a cube is made of six identical square surfaces. First, find the area of one face: 5 × 5 = 25. Then, multiply by 6 to account for all of the faces: 25 × 6 = 150.
The figure here shows two wooden cubes joined to form a rectangular solid. If each cube has a surface area of 24, what is the surface area of the resulting rectangular solid?