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# Surface Area and Volume

# Surface Area of Prisms and Cylinders

We've calculated the lateral area for both prisms and cylinders. Lateral area is part of surface area, but it doesn't tell the whole story. We need the bases, too.

No, not those bases.

The surface area *SA* equals the lateral area *L* plus the area of the two bases. Since the bases are congruent, they're identical. In Mathspeak,

*SA* = *L* + 2*B*

The ways to calculate *L* and *B* for cylinders and prisms are as different as night and later that night. We'll go through it and you'll see what we mean...hopefully.

### Sample Problem

This is a trapezoidal prism. Sounds more like a disease than a shape ("I have the trapezoidal prism flu!"). What is the surface area of the prism?

We can find the surface area with all the formulas we know.

*SA* = *L* + 2*B*

We'll start with the lateral area, which is basically a big rectangle. We know that the area of a rectangle is *bh* and in this case, our *b* is really a *P* (the perimeter of the base). So our first step is to find *P*. Some say it's right before Q, but those might just be rumors.

We're missing one crucial side of *P*: the hypotenuse of the right triangle. Thankfully, we know the other two sides and the Pythagorean theorem. Have at it.

*a*^{2} + *b*^{2} = *c*^{2}

6^{2} + (8 – 5)^{2} = *c*^{2}

36 + 9 = *c*^{2}

45 = *c*^{2}*c* ≈ 6.7

Hooray. Now we can find *P*, which is just the perimeter of the trapezoid.

*P* = 8 + 6 + 6.7 + 5 *P* = 25.7

We know our height, so we can find the lateral area.

*L* = *Ph* = (25.7)(12) = 308.4

We have *part* of the surface area. Now let's get back to the base-ics.

The trapezoid is just a rectangle and a triangle together. If we find the area of both and add them, we'll get the area of the base.

The base of the rectangle is 5 and the base of the triangle is 3. The height of both is 6. Plug it in, plug it in.

Finally, we can find our surface area.

*SA* = *L* + 2*B**SA* = 308.4 + 2(39) = 386.4 units^{2}

Whew. We didn't even need chicken noodle soup beat the trapezoidal prism flu. We got through it all on our own.

What about the surface area of cylinders? If you ever want to gift-wrap some toilet paper (after the apocalypse, when rainforests are gone and paper is more expensive than diamonds), you'll need to know this one. Again, we'll start with the generic surface area formula.

*SA* = *L* + 2*B*

We know the lateral surface fsaarea is a triangle with a height equal to the altitude and a width equal to the circumference.

*L* = 2π*rh*

Also, the bases are circles, with areas equal to π*r*^{2}. Putting that all together, we have:

*SA* = 2π*rh* + 2(π*r*^{2})

Let's mix it up a little. Here's a video explaining the surface area of a cube. It happens to talk about the volume of a cube as well just to further enrich those brains out there.

### Sample Problem

What is the surface area of this cylinder?

We know the height and the radius, so we can just substitute them into the formula.

*SA* = 2π(5 in)(8 in) + 2π(5 in)^{2}*SA* = 80π + 50π = 130π = 408.4 in^{2}

Congratulations. Now you can give your friends the gift of hygiene in a post-apocalyptic world. Of course, buying the wrapping paper might be just as expensive as buying the toilet paper itself.