A rectangular prism is the name for an object with 6 sides that everyone is very familiar with – a square. Think of a brick or a shoebox, that's a perfect example of a rectangular prism. Surface area is the sum of the surface areas of an object. "How much paper do I need to wrap this shoebox?" sounds simpler, but it's also a matter of math.
Step
Part 1 of 2: Finding Surface Area
Step 1. Label the length, width, and height
Each rectangular prism has a length, width, and height. Draw a prism, and write the symbols p, l, and t next to three different sides of the wake.
- If you're not sure which side to label, select any corner point. Label the three lines that meet at that vertex.
- For example: A box has bases 3 meters and 4 meters long and stands 5 meters high. The side length of the base is 4 meters, so p = 4, l = 3, and t = 5.
Step 2. Look at the six sides of the prism
To cover the entire large surface, you'll need to paint six different sides. Imagine one at a time – or find a cereal box and see it in person:
- There are ups and downs. Both are the same size.
- There are front and back sides. Both are the same size.
- There are left and right sides. Both are the same size.
- If you're having trouble imagining it, cut a square along the edges and spread it out.
Step 3. Find the area of the bottom side
To start, let's find the surface area of one side: the bottom. This side is a rectangle, just like all the sides. One side of the rectangle is labeled length and the other side is labeled width. To find the area of a rectangle, just multiply the two edges. Area (bottom side) = length times width = pl.
Returning to our example, the area of the bottom side is 4 meters x 3 meters = 12 meters squared
Step 4. Find the area of the top side
Wait – we already know that the top and bottom sides are the same size. The top side must also have an area pl.
In our example, the top area is also 12 square meters
Step 5. Find the area of the front and back sides
Go back to your diagram and look at the front side: the side with one edge labeled width and one edge labeled height. Front side area = width times height = lt. The area of the back side is also lt.
In our example, l = 3 meters and t = 5 meters, so the area of the front side is 3 meters x 5 meters = 15 meters squared. The area of the back side is also 15 square meters
Step 6. Find the area of the left and right sides
We only have two sides left, both of which are the same size. One edge is the length of the prism, and the other edge is the height of the prism. The area of the left side is pt and the area of the right side is also pt.
In our example, p = 4 meters and t = 5 meters, so the area of the left side = 4 meters x 5 meters = 20 meters squared. The area of the right side is also 20 square meters
Step 7. Add up the six areas
Now, you have found the area of the six sides. Add up the areas to get the total area of the figure: pl + pl + lt + lt + pt + pt. You can use this formula for any rectangular prism, and you'll always get the surface area.
To complete our example, just add up all the blue numbers above: 12 + 12 + 15 + 15 + 20 + 20 = 94 square meters
Part 2 of 2: Simplifying Formulas
Step 1. Simplify the formula
Now you know enough about how 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 = pl + pl + lt + lt + pt + pt. If we combine all the equal terms, we get:
Area of rectangular prism = 2pl + 2lt + 2pt
Step 2. Factor out the number two
If you know how to factor in algebra, you can simplify the formula:
Area of Rectangular Prism = 2pl + 2lt + 2pt = 2(pl + lt + pt).
Step 3. Test the formula in the example
Let's go back to our example box, with a length of 4, a width of 3, and a height of 5. Plug these numbers into the formula:
Area = 2(pl + lt + pt) = 2 x (pl + lt + pt) = 2 x (4x3 + 3x5 + 4x5) = 2 x (12 + 15 + 20) = 2 x (47) = 94 square meters. This is the same answer, which we got earlier. Once you practice doing these equations, this formula is a much faster way to find surface area
Tips
- Area always uses squared or square units, such as square meters or square centimeters. A square meter, as the name implies, is: a square that is one meter wide and one meter long. If a prism has an outer surface of 50 square meters, it means that we need 50 squares to cover the entire surface of the prism.
- Some teachers use depth instead of height. This term is fine, as long as you label each side clearly.
- If you don't know which part is the top of the prism, you can call any side a height. Length is usually the longest side, but it doesn't really matter. As long as you use the same names in all the questions, you shouldn't have any problems.
