# Volume & Surface Area of a Truncated Cone

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• 0:04 Truncated Cone Definition
• 1:12 Truncated Cone Volume
• 2:42 Truncated Cone Surface Area
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Lesson Transcript
Instructor: Laura Pennington

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

In this lesson, we'll learn what a truncated cone is and what one looks like. We'll then go on to see the formulas we can use to calculate the volume and surface area of a truncated cone as well as apply them to an example.

## Truncated Cone Definition

What was the last really great film you went to see at a movie theater? Going to the movies is always a great time! Of course, the experience of going to a movie at the theater isn't complete without popcorn! Most likely, you've never taken note of the shape of a popcorn tub at the theater, but here's an interesting fact! There is a mathematical name for this shape, and it's called a truncated cone.

A truncated cone is a three-dimensional object that looks like a cone with the top pointy end chopped off.

The two circular ends of a truncated cone are called the bases. We will call the radius of the smaller circular base the little radius and represent it with r, and we will call the radius of the larger circular base the big radius and represent it with R. The distance between the centers of the two bases is called the height of the truncated cone, and we represent it with h. Lastly, the shortest distance between the outer edges of the bases is called the slant height, and we represent it with s.

Do you now see that a popcorn tub has this shape? Pretty interesting, huh? Well, it's about to get even more interesting! Let's talk a bit about the volume and surface area of a truncated cone!

## Truncated Cone Volume

Suppose you go to the movies, and you get a large popcorn. You can't wait to sink your teeth into the salty, buttery goodness, but before doing so, you begin to wonder just how much popcorn fits inside the large popcorn tub. I've got news for you! You're actually wondering what the volume of the popcorn tub is!

The volume of a truncated cone is the amount of space inside the truncated cone. The good news is that we have a nice formula that we can use to find the volume of a truncated cone.

Volume = (1/3) â‹… Ï€ â‹… h(R 2 + r 2 + R â‹… r)

Your curiosity gets the best of you, so you ask the worker at the concessions counter to see an empty large popcorn tub. When he gives it to you, you take some quick measurements and find that the big radius is 5 inches, the small radius is 4 inches, the height is 10 inches, and the slant height is 11 inches. You give the empty tub back to the concessions worker and pull out a pencil and some paper to calculate the volume of the large popcorn tub using your formula.

So, we see that the volume of the large popcorn tub is approximately 638.79 cubic inches. Wow! That's a lot of popcorn! Dig in!

## Truncated Cone Surface Area

As you begin to gobble down your popcorn, you begin to have a bit of buyer's remorse. After all, you paid \$8 for this large popcorn, and that's not cheap! This gets you to wondering just how much the movie theater has to pay for a popcorn tub. You have a bit more time until your movie starts, so you ask a worker how much these tubs cost them. The worker says that the manufacturer charges based on how much material was used to make the tub and a lid for the tub, and they charge 0.00125 cents per square inch of material used.

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