Surface Area

So, now let's talk about surface area.  It's exactly that, the area of the surface of a 3D shape.  So if you understand how the 3D Net works of that shape, the surface area makes sense.

The general formula for surface area for prisms is as follows:

The perimeter of the base times the height, plus the area of the base times two.  So, let's look at a few examples for more information.

Here is a triangular prism.  It's 3D net is similar to this:

The large rectangle in the middle has a base equal to the height of the shape, and a height equal to the perimeter of the base.

So let's follow the same steps we use for Volume and work this out.

1. What shape is it?  Already identified.

2. Write our formula.

General                              S.A. = P * h + 2 * B

Specific                              S.A. = (s+s+s) * h + 2 * (0.5 * b * h)

Remember that the height in the general formula refers to the height of the prism.  Also remember to follow order of operations.  The area of the base is still the area of a triangle formula, just like with volume.

3. Plug in the numbers

                                        S.A. = (9+8+7) * 5 + 2 (0.5 * 9 * 6)

That looks like a lot, but with practice, it will get easier.

4. Solve

                                       S.A. = (24) * 5 + 2 (27)

                                       S.A. = 120 + 54

                                       S.A. = 174 square miles

Since it's area, it won't be cubic miles.  We're measuring the covering on the outside of the shape, not how much space is contained inside it.  Let's try another.

1. What shape is it?

This is a cylinder.  It's net looks like: 

2. Write the formula

Same general formula:              S.A. = P * h + 2 * B

Different specific formula:         S.A. = (2 * pi * r) * h + 2 * (pi * r^2)

Since we're dealing with circles, we use circumference for perimeter and area of a circle for our base.

3. Plug in the numbers
diameter not radius was given   S.A. = (2 * 3.14 * 10) * 5 + 2 * (3.14 * 10^2)
10^2 = 100                             S.A. =  (62.8) * 5 + 2 * (314)
                                               S.A. = 314 + 628
                                               S.A. = 942 square inches.

Last example for now:

1. What shape is it?

This is a rectangular prism.  Never mind the slant, it's still a rectangular prism.

Pick bases:  I'm picking the 3 x 5 sides, as they'll be easy to use.

2. Write formulas:

Same general formula:              S.A. = P * h + 2 * B

Specific formula:                      S.A. = (b+b+h+h) * h + 2 * (b * h)

Again, note that the h outside parenthesis refers to the height of the prism.

3. Plug in the numbers

                                              S.A. = (3+3+5+5) * 12 + 2 * (3 * 5)

4. Solve

                                              S.A. = (16) * 12 + 2 * (15)

                                              S.A. = 192 + 30

                                              S.A. = 222 square inches