A rectangular prism is a name for a 6 sided 3 dimensional figure that is very familiar to everybody a box. Think of a brick or a shoebox and you know exactly what a rectangular prism is. The surface area is the amount of space on the outside of the object. How much paper do I need to wrap this shoebox sounds a lot less complicated but it's exactly the same math problem. The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one dimensional curve or of the surface area for polyhedral i.e. objects with flat polygonal faces for which the surface area is the sum of the areas of its faces. Smooth surfaces such as a sphere are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. Each rectangular prism has a length a width and a height. Draw a picture of the prism and write the symbols l’ w and h next to three different edges of the shape. If you're not sure which sides to label pick any corner? Label the three lines that meet at that corner. For example a box has a base that measures 3 inches by 4 inches and it stands 5 inches tall. The long side of the base is 4 inches so l = 4’ w = 3’ and h = 5. To cover the whole surface area you'd need to paint six different faces. Think about each one or find a box of cereal and look at them directly. There is a top and bottom face. Both are the same size. There are a front and a back face. Both are the same size. There are a left and right face. Both are the same size. If you have trouble picturing this cut a box apart along the edges and lay it out. To start out let's find the surface area of just one face the bottom. This is a rectangle just like every face. One edge of the rectangle is labeled length and the other is labeled width. To find the area of the rectangle just multiply the two edges together. Area bottom edge = length times width = law. Going back to our example the area of the bottom face is 4 inches x 3 inches = 12 square inches . Wait a second we already noticed that the top and bottom faces are the same size. This must also have an area of LW. In our example the top area is also 12 square inches . Go back to your diagram and look at the front face the one with one edge labeled width and one labeled height. The area of the front face = width times height = WH. The area of the back is also WH. In our example w = 3 inches and h = 5 inches so the area of the front is 3 inches x 5 inches = 15 square inches . The area of the back face is also 15 square inches . We've just got two faces left each the same size. One edge is the length of the prism and one edge is the height of the prism. The area of the left face is lh and the area of the right face is also lh. In our example l = 4 inches and h = 5 inches so the area of the left face = 4 inches x 5 inches = 20 square inches . The area of the right face is also 20 square inches . Now you've found the area of each of the six faces. Add them all together to get the area of the whole shape LW + LW + WH + WH + LH + LH. You can use this formula for any rectangular prism and you will always get the surface area. To finish our example, just add up all the blue numbers above: 12 + 12 + 15 + 15 + 20 + 20 = 94 square inches . You now know enough to find the surface area of any rectangular prism. You can do it faster if you've learned some basic algebra. Start with our equation above: Area of a rectangular prism = LW + LW + WH + WH + LH + LH. If we combine all the terms that are the same we get. Area of a rectangular prism = 2lw + 2wh + 2lh If you know how to factor in algebra you can make it even shorter. Area of a Rectangular Prism = 2lw + 2wh + 2lh = 2 LW + WH + LH. Let's go back to our example box with length 4 width 3 and height 5. Plug these numbers into the formula. Area = 2(lw + wh + lh) = 2 x (lw + wh + lh) = 2 x (4x3 + 3x5 + 4x5) = 2 x (12 + 15 + 20) = 2 x (47) = 94 square inches. That's the same answer we got before. Once you've practiced doing these equations this is a much faster way to find the surface area.