Surface Area and Volume
Topics
Introduction to :
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 + 2B
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 + 2B
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 baseics.
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 + 2B
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 giftwrap 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 + 2B
We know the lateral surface fsaarea is a triangle with a height equal to the altitude and a width equal to the circumference.
L = 2πrh
Also, the bases are circles, with areas equal to πr^{2}. Putting that all together, we have:
SA = 2πrh + 2(πr^{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 postapocalyptic world. Of course, buying the wrapping paper might be just as expensive as buying the toilet paper itself.
Example 1
Find the lateral area and the surface area of the solid. 
Example 2
Find the lateral area and the surface area of the cylinder. 
Example 3
The lateral area of a regular hexagonal prism is 81 ft^{2}. If the prism has a height of 3 ft, what is the length of one side of the hexagonal base? 
Example 4
Find the lateral and surface areas of the solid.

Exercise 1
Find the total surface area of the solid.
Exercise 2
Find the missing dimensions of the triangular prism if the surface area is 912.1 square inches and the lateral area is 800.1 square inches.
Exercise 3
Find the lateral and surface area of the cylinder.
Exercise 4
Find the lateral and surface area of the trapezoidal prism.
Exercise 5
Find the radius and circumference of the circle at the base of the cylinder if the lateral area is 150 yd^{2}. Then find the surface area.
Exercise 6
A regular dodecagon (12 sides) is the base of a prism of height 19 inches. If the lateral area of the prism is 228 square inches, what is the length of each side of the dodecagon?
Exercise 7
Find the lateral and surface area of the triangular prism.
Exercise 8
After the apocalyptic destruction of the forests, only one tree stump remains. You want to preserve it, so you decide to cover it in plastic wrap. What better way to preserve a dead tree? Assuming the tree is perfectly cylindrical with a height of 3 feet and a diameter of 2.4 feet, find out how much plastic wrap will you need to cover the top and sides of the tree.
Exercise 9
The Catfish Factory has gone out of business due to too much cat hair in their tuna. As a result, all the cat workers are homeless. Since these cats are so darn cute (and you're tired of them pooping on your driveway), you decide to buy them a big litter box. How much paper will you need to giftwrap their new litter box?