Surface Area In this lesson, we'll learn what surface area is, and how to figure out the surface area of rectangular solids and cylinders.
Look at the rectangular solid at right. Sometimes it is called a rectangular prism, but you can just think of it as a box. Let's say we want to paint the box, and we want to know how much surface we'll actually be painting.
What we need to know is the area of each of the six sides. Each side is a rectangle, and we already know how to find the area of a rectangle. We just multiply the two dimensions. Let's look at how we can figure out the surface area, which means the area of all six sides combined. The area of the yellow side is L·H. Where else is there a yellow side on this box? The side opposite (which we can't see), has the same dimensions. That means we'll be computing L·H twice. This can be written as 2LH.
The area of the blue side is D·H, which are the two dimensions involved. Again, there is another blue side opposite which we can't see, but we need to add the two of them, so we have 2DH. The orange side is a bit harder to visualize, but the area is L·D. Again, there are two of them, so we have 2LD.
Therefore, the surface area of a rectangular solid is given by: A = 2LD + 2LH + 2DH
Let's look at the cylinder at right. We're assuming that it is not hollow. We'll need to paint the length of the tube, as well as the two circular ends. What the area of each circular end? Recall that the area of a circle is πr2. We're adding two of those circles, so we have 2πr2.
Now we have to figure out the area of the tube. What is the distance around the tube? It's circular, so we need to figure out the circumference. Our formula for that is 2πr. That's the distance around, but now we need to multiply it by the height of the cylinder. Let's call it h. That means the surface area of the tube is 2πrh. We need to remember to add in the area of the two circular ends, so our surface area formula becomes: A = 2πr2 + 2πrh
These formulas can be hard to memorize. You should try to, but what is much more important is to understand where they come from, and make sure that they make sense to you. If they do, it will be very easy for you to recreate them when you need to.
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