# Surface Area of a Tube

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

You might know what a tube is, but in this lesson, we will learn about it mathematically. We will learn what the surface area of a tube is and how to find that surface area using a formula - all while rescuing a fuzzy little kitten!

## The Tube

Suppose you are having some construction work done in your backyard. One day, you head out back to see how things are coming along, and you hear something coming from the area where the construction equipment is located. You walk over to investigate, and lo and behold, you look into one of the concrete tubes and see this staring back out at you.

It's a stray kitten! Wow! Is it ever adorable! You walk over to it, and discover that it's extremely friendly and lovable. You realize that this is going to be your new pet as you instantly fall in love with it. You try to lure it out of the concrete tube to come inside so you can clean it up and feed it, but every time it comes out, it runs back in as if the tube is its safe place. You desperately want to rescue this little fluff ball, so you decide to just move the whole concrete tube into your living room so the little guy can have it's safety net as it gets used to its new home. You ask one of the construction workers to help you, and you two manage to get the tube inside (kitty included!). You and kitty settle in and start to get to know each other. This is the start of a beautiful friendship!

As it turns out, your kitty's little safe place is actually a mathematical object called a tube. A tube is a three dimensional object that is shaped like a cylinder, but has a hole through the middle of it. We call the radius of the hole through the middle the small radius, and we call the radius of the whole tube the big radius. The length of the tube is called the height.

## Surface Area of a Tube

Okay, it's been a few days with kitty now, and things are going great. Kitty loves snuggling and playing with you. However, you've found that it loves to sleep in the tube, because it feels safe there. You want to make your new family member as comfortable as possible, so you come up with the brilliant idea of covering the whole tube, inside and out, with a thick foam fabric. This will give kitty some cushioning while sleeping and also make it possible for it to use the tube as a bit of a jungle gym that it can climb on.

You are about to head to the store to get the foam fabric when you realize you don't know how much you need. This puts you in a pickle - somehow you need to figure out how much area is taken up on the entire surface of the tube. Well, here's some great news: there is a name for this exact characteristic of a tube, and it's called the tube's surface area. The surface area of a tube is the amount of area that the tube's surface takes up (inside and out). Now try to contain your excitement, because there's even better news! We have a formula to calculate a tube's surface area, so you will be able to figure out the exact amount of foam fabric you will need!

To find the surface area of a tube, we let R = the big radius, r = the small radius, and h = the height of the tube, and we use the following formula.

• Surface Area = 2π(R 2 - r 2) + 2πh(R + r)

Great! Let's use this formula to figure out how much foam fabric you will need for kitty's safe haven.

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