In this lesson, we'll learn about the volume formulas for cylinders, cones and spheres. We'll also practice using the formula in a variety of real-world examples where knowing how to calculate volume is helpful.
What Is Volume?
Have you ever watched an eating contest? It can be kind of gross. But, what always amazes me is when seemingly skinny people win. How can someone fit so much food into such a tiny stomach? It's a question of volume. Volume is the capacity of an object, or how much space it occupies. An eating contest tests the volume of the human stomach. In this lesson, we're not going to test the limits of anyone's stomach, but we are going to learn about the volume formulas for some common shapes.
A cylinder is basically like a big pile of circles. Imagine you have a poker chip. Then, you start winning, and you have a bunch of chips. If you stack them together, you've created a cylinder. This should help you remember the volume formula.
The volume of a cylinder is pi*r^2*h, where r is the radius of the circle on the cylinder's end. What looks familiar in there? Pi*r^2 - that's the area of a circle. And, if you had just one perfectly flat poker chip, its area would be pi*r^2. But, since you're a baller, you have a stack. So, you take the area formula, and you multiply it by the height of your chips.
Let's try some examples. Below is a can of a new soda, MegaSurge. Not only is the neon yellow soda packed with 10 times the caffeine of regular soda, it also comes in an oversized can. But, how much soda does it hold? You measure the width of the top, which is the diameter of the circle, and it's 6 inches. So the radius, which is half the diameter, is 3 inches. And, how tall is it? 9 inches. Let's use our volume formula: pi*r^2*h. Pi*3^2*9. That's about 254 cubic inches. That's a lot of soda!
You can use the volume formula for a cylinder to find the volume for this can of soda.
Okay, here's another. After drinking that soda, you decide you need to get back to more healthy foods, so you visit your aunt's farm for some fresh veggies, eggs and other good stuff. She has a grain silo. You want to know how tall it is. She tells you it holds 3,141 cubic feet of grain when full. So, you know its volume. And, the radius is 5 feet. How tall is it? Let's use our formula! 3141 = pi*5^2*x. 3141 = 79x. x = 40, so it's about 40 feet tall.
As it turns out, your aunt uses that grain to feed her cows, and she uses her cows' milk to make ice cream. Your aunt is kind of awesome. And, now you want to know how much ice cream you can pack into your cone. The more ice cream you can pack into the cone, the happier you'll be if the scoop on top should fall off or, you know, gets eaten too quickly.
The volume of a cone is 1/3*pi*r^2*h, where r is the radius of the circle at the wide end of the cone. Notice how this is the same as the cylinder formula. There's just that extra 1/3. What's that for? Well, one end is a circle and one is a point. So, there is 1/3 of the volume of a cylinder. In other words, you'd need 3 cone-shaped cones to match the volume of 1 cylinder-shaped cone. Maybe those people with the cake cones are onto something.
But, let's talk about the cone you have. It's 5 inches tall and 2 inches across the top. Remember, that 2 inches is the diameter. That means the radius is 1 inch. Okay, the formula: 1/3*pi*r^2*h. That's 1/3*pi*1^2*5, which equals a little over 5 cubic inches. That's not a lot of ice cream.
So, you convince your aunt to make waffle cones, which are bigger, but how much bigger? You make a waffle cone that's 8 inches tall and 5 inches across the top. So, its radius is 2.5 inches. What is the volume it can hold? 1/3*pi*r^2*h. That's 1/3*pi*2.5^2*8, or just over 52 cubic inches. That's about 10 times the ice cream! That's a big waffle cone.
Maybe you should get some exercise after all that ice cream and play some basketball. You decide to impress your friends with your awesome geometry skills by explaining how much air fits in the basketball. But wait, the basketball is a sphere. What's the formula for that?
The volume of a sphere is 4/3*pi*r^3, where r is the radius of the sphere. Where does this come from? Well, I could just tell you 'calculus,' but that wouldn't help you remember it. But, if you remember that the surface area of a sphere is 4*pi*r^2, you can remember that volume is related to surface area. The other volume formulas we've been discussing involve 3s, so if you add some 3s to the surface area formula, you get 4/3*pi*r^3.
That 4/3 is definitely unusual, which is the opposite of spheres. Spheres are everywhere! They're as small as the tiny peas from the garden and as huge as the Death Star in Star Wars. Without spheres, there'd be no baseball, soccer, tennis or golf. Well, there'd be no Earth either, which is kind of important.
But, back to the basketball. All you need to know is its radius or diameter. You know the diameter is 9 inches, so its radius is 4.5 inches. To the formula! 4/3*pi*r^3. 4/3*pi*4.5^3. That's about 382 cubic inches.
Your friends are impressed. In fact, one asks you to help him with his science project. He's building a model of the solar system, and he's trying to understand the difference in sizes between the planets. Your ice cream experiment makes you think that volume may be a good way to help.
He knows the diameter of Earth is about 7,900 miles, the diameter of Mars is about 4,200 miles, and the diameter of Jupiter is about 89,000 miles. But, what does that mean in volume? Okay, first, let's convert those all to radii. Earth's radius is 3,950 miles. With Mars, it's 2,100 miles. And, Jupiter's is 44,500 miles. Let's break out that volume formula: 4/3*pi*r^3.
First, Earth: 4/3*pi*3950^3 = 258 billion cubic miles. That's a lot!
Next, Mars: 4/3*pi*2100^3 = 39 billion cubic miles. That's a lot smaller.
What about Jupiter? 4/3*pi*44,500^3 = 369 trillion cubic miles! That means well over 1,000 Earths could fit inside Jupiter! Your buddy is going to need a bigger model.
We learned about 3 unique shapes. First, cylinders, like that can of MegaSurge or the silo. The formula for the volume of a cylinder is pi*r^2*h. Second, we looked at cones as we feasted on ice cream. The formula for the volume of a cone is 1/3*pi*r^2*h. Finally, we looked at spheres, like a basketball and the planets. The formula for the volume of a sphere is 4/3*pi*r^3.
After reviewing this lesson, you'll have the ability to:
- Identify the volume formulas for cylinders, cones and spheres
- Solve real-world example problems using these formulas
- Explain why the volume formulas for these three shapes are different