## Surface area of rectangular prisms

 To measure the area of a 2-dimensional object, you need to count the number of unit squares necessary for covering the object. A unit square can be 1" x 1" or 1 yd x 1 yd or 1 ft x 1 ft or a square by some other unit. To find the area of a 3 cm-by-5 cm rectangle, you would imagine that rectangle cut into unit squares. Then you would count the squares. There are 15 unit squares, so the area is \$15\mbox{ cm}^2\$. Of course, a short cut for finding the area of a rectangle is to multiply the length by the width. A = length • width A = lw

For the surface area of a solid, there is a similar definition, but it applies to the exterior surfaces of the solid. Surface area is the sum of all unit squares that fit on the exterior of a solid.

As the diagram below indicates, there are six surfaces to a rectangular prism. There is a front, back, top, bottom, left, and right to every rectangular prism. The surface are of a prism is nothing more than the sum of all the areas of these rectangles.

Notice that the top and bottom faces will have equal areas. The front and back faces will have equal areas. Finally, the left and right faces will have equal areas. This means instead of finding the area of all six faces, we only need to find the areas of three faces, multiply each by 2, and then find their sum.

Surface Area = Top + Bottom + Front + Back + Left + Right

Surface Area = 2 x Top + 2 x Front + 2 x Left

Surface Area = 2lw + 2lh + 2wh

 Surface area of rectangular prism Surface Area = \$2lw+2lh+2wh\$

 Example 1 Find the surface area for this rectangular prism. SA = 2 • Top + 2 • Front + 2 • Left SA = 2lw + 2lh + 2wh SA = 2 • 4 • 3 + 2 • 4 • 5 + 2 • 3 • 5 SA = 24 + 40 + 30 SA = \$94\mbox{ cm}^2\$

 Example 2 Find the surface area for this rectangular prism. SA = 2 • Top + 2 • Front + 2 • Left SA = 2lw + 2lh + 2wh SA = 2 • 6 • 3 + 2 • 6 • 3 + 2 • 3 • 3 SA = 36 + 36 + 18 SA = \$90\mbox{ cm}^2\$

 Example 3 Find the surface area for this rectangular prism. SA = 2 • Top + 2 • Front + 2 • Left SA = 2lw + 2lh + 2wh SA = 2 • 5 • 4 + 2 • 5 • 7 + 2 • 4 • 7 SA = 40 + 70 + 56 SA = \$166\mbox{ cm}^2\$

Geogebra applet:

Use the length, width, and height sliders to create a rectangular prism of your choosing. Notice how the net on the right changes as the move the sliders.

# Self-Check

Question 1

Find the surface area for this rectangular prism.

Question 2

Find the surface area for this rectangular prism.

Question 3

Find the surface area for this rectangular prism.