Aquarium Splash

Surface Area

To find the surface are of this square prism (square prism because the bases are squares), we can find the area of each surface by breaking the prism into a net.

Define each dimension:

b = 4 inw = 4 inh = 4 in

b x w+ b x w + b x h + b x h+ b x h + b x h
(4 in x 4 in) + (4 in x 4 in) + (4 in x 10 in) + (4 in x 10 in) + (4 in x 10 in) + (4 in x 10 in) =
16 in2 + 16 in2 + 40 in2+ 40 in2 + 40 in2 + 40 in2 = 192 in2

If we take a look at the similar areas that we’ve found, the 2 congruent bases and the 4 congruent faces, we can use some repeated math for which we may be able to create a shortcut.

In order to make math easier down the road, we can simplify formulas by eliminating some defined variables. For instance, since we are working with three dimensions in a prism, we can refer to the base area (b x w) as B. From here on out, we will refer to the base area as B.

Since we’ve found the area of the square bases twice, instead of calculating the area for each base, we can find the area of one base and multiply by two. So the area of the two bases of the prism will be 2B.

Since we’ve found the area of the rectangular faces four times, instead of calculating the area for each face, we can see the repeated reasoning (MP 8.)

(4 in x 10 in) + (4 in x 10 in) + (4 in x 10 in) + (4 in x 10 in)

If we look at each of these four terms, we can see two common factors; 4 inches and 10 inches. The 10 inches represents the height of each face of the prism. It is the same for each of the four faces. If we factor out 10 in from each term in the expression above, it looks like this:

10 in (4 in + 4 in + 4 in + 4 in)

Another way to represent adding all the sides of a shape together is P for Perimeter.

We can multiply the height of our prism by the Perimeter of the prism base, and that will provide the area for all the faces of the prism. Ph

So by finding the sum of the areas of both bases (2B) and the areas of all the faces (Ph), we have a consistent formula for finding the surface area of any prism.

You will get the same surface area if you choose to find the area of each surface individually, but “ain’t nobody got time for that!”

Additional Examples:

It is clear that the bases of this prism are trapezoids and the faces are rectangles. So to find the area of the bases we use the formula for the area of a trapezoid:

If the dimensions for the trapezoid are b1 = 5 cm, b2= 7 cm, h = 2 cm and s = 6 cm, and the height of the prism is 11 cm, then we start by finding the base area:

B =12 cm2

Then we find the perimeter of the base of the prism:

Triangular Prism: The height of the prism’s base is different from the height of the prism.
The dimensions of the triangular bases are: b = 14 m, h = 6 m, s = 8 m.
The height of the prism is 12 m.