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# 10.15: Volume of Cones

Difficulty Level: At Grade Created by: CK-12
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Candice and Trevor have been working together for two weeks are getting to be like siblings. Every day there seems to be an argument of some kind that erupts between the two of them. Today, it was about ice cream cones.

“I think that a plain cone with a pointed bottom holds more ice cream,” Trevor said reaching for more tape.

“I don’t. A sugar cone with a pointed bottom definitely holds more.”

“We can solve this easily if we had the dimensions.”

“Well, on your break walk down to the ice cream parlor and figure this out,” Candice instructed.

So on his break Trevor did exactly that. He walked down to the ice cream corner and came back with some dimensions on a piece of paper.

“Here we go, now you figure out the volume of the sugar cone and I’ll do the plain one.”

Here are the dimensions that they have to work with:

Plain Cone =H=3.25′′ D=2.5′′\begin{align*}= H = 3.25'' \ D = 2.5''\end{align*}

Sugar Cone =H=4.5′′ D=2′′\begin{align*}= H = 4.5'' \ D = 2''\end{align*}

The two got right down to work.

Now it’s your turn. You will learn all about the volume of cones and pyramids in this Concept. By the end of it, you will know which cone holds the most ice cream.

### Guidance

To figure out the volume of a cone, let’s first look at how we can compare it with another solid figure. The closest solid figure to a cone is a cylinder. Think about it. Both of them have circular bases. While a cylinder has two circular bases, a cone only has one. But they are both figures that we can compare.

Here is a cylinder and the formula for finding the volume of a cylinder.

V=πr2h\begin{align*}V = \pi r^2 h\end{align*}

Now let’s think about a cone. How can we find the volume of a cone using the information that we have learned about cylinders? We know that they are related, but is there a way to see how the volume of the cylinder can be compared to the volume of a cone? What about if we put the cone inside the cylinder?

Take a look.

Now you can see how they are related. Notice that the base is the same in both the cone and the cylinder. Think back to the prism and the pyramid!

To find the volume of the cone, we are going to use a formula similar to that of a pyramid, except that we are going to take into consideration that the base of a cone is a circle. Therefore, we will need to find the area of the circle to find the volume of the cone.

V=13πr2h\begin{align*}V=\frac{1}{3} \pi r^2 h \end{align*}

Here we need to find the area of the circle that is the πr2\begin{align*}\pi r^2\end{align*}. Then we can take that measurement and multiply it by the height of the cone. Next, because the cone fills a portion of the cylinder, we can take one-third of the product.

Take a few minutes and copy this formula down in your notebook.

Now that you have some understanding about the volume of a cone and about where the formula comes from, let’s practice using it to figure out the volume of a cone.

Find the volume of the following cone.

We can start by substituting the given values into the formula.

VVVVV=13πr2h=13(3.14)(52)(7)=13(3.14)(175)=13(549.5)=183.16 in3\begin{align*}V & = \frac{1}{3} \pi r^2 h \\ V & = \frac{1}{3} (3.14)(5^2 )(7) \\ V & = \frac{1}{3} (3.14)(175) \\ V & = \frac{1}{3}(549.5) \\ V & = 183.16 \ in^3\end{align*}

This is our answer. Notice that we rounded it to the nearest hundredth.

Here is another one.

What is the volume of the cone below?

First, we can substitute the given values into the formula.

VVVV=13(3.14)(32)12=13(3.14)(108)=13(339.12)=113.04\begin{align*}V & = \frac{1}{3} (3.14)(3^2)12 \\ V & = \frac{1}{3} (3.14)(108) \\ V & = \frac{1}{3}(339.12) \\ V & = 113.04\end{align*}

The volume of the cone is 113.04 cm3\begin{align*}113.04 \ cm^3\end{align*}.

Find the volume of each cone using the given values and the formula.

#### Example A

A cone with a radius of 2 inches and a height of 4 inches. You may round to the nearest hundredth as needed.

Solution: 16.74 in3\begin{align*}16.74 \ in^3\end{align*}

#### Example B

A cone with a radius of 5 ft and a height of 8 ft.

Solution: 209.33 ft3\begin{align*}209.33 \ ft^3\end{align*}

#### Example C

A cone with a radius of 4 m and a height of 9 m.

Solution: 150.72 m3\begin{align*}150.72 \ m^3\end{align*}

Here is the original problem once again.

Candice and Trevor have been working together for two weeks are getting to be like siblings. Every day there seems to be an argument of some kind that erupts between the two of them. Today, it was about ice cream cones.

“I think that a plain cone with a pointed bottom holds more ice cream,” Trevor said reaching for more tape.

“I don’t. A sugar cone with a pointed bottom definitely holds more.”

“We can solve this easily if we had the dimensions.”

“Well, on your break walk down to the ice cream parlor and figure this out,” Candice instructed.

So on his break Trevor did exactly that. He walked down to the ice cream corner and came back with some dimensions on a piece of paper.

“Here we go, now you figure out the volume of the sugar cone and I’ll do the plain one.”

Here are the dimensions that they have to work with:

Plain Cone =H=3.25′′ D=2.5′′\begin{align*}= H = 3.25'' \ D = 2.5''\end{align*}

Sugar Cone =H=4.5′′ D=2′′\begin{align*}= H = 4.5'' \ D = 2''\end{align*}

The two got right down to work.

Trevor started with the plain cone. Here is the formula that he used.

VVV=13πr2h=13(3.14)(1.252)(3.25)=5.315 or 5.3 cubic inches\begin{align*}V & = \frac{1}{3} \pi r^2 h \\ V & = \frac{1}{3} (3.14)(1.25^2)(3.25) \\ V & = 5.315 \ \text{or} \ 5.3 \ cubic \ inches \end{align*}

Candice worked on the sugar cone.

VVV=13πr2h=13(3.14)(12)(4.5)=4.71 or 4.7 cubic inches\begin{align*}V & = \frac{1}{3} \pi r^2 h \\ V & = \frac{1}{3} (3.14)(1^2)(4.5) \\ V & = 4.71 \ \text{or} \ 4.7 \ cubic \ inches \end{align*}

While both cones are very close in size, the plain cone is a little greater in size, so that one would hold more ice cream.

### Vocabulary

Here are the vocabulary words in this Concept.

Pyramid
a solid figure with a polygon as a base where the sides meet in one single vertex at the top.
Cone
a solid figure with a circular base where the sides are rounded yet meet at one vertex at the top.
Volume
the amount of space contained inside a solid figure.

### Guided Practice

Here is one for you to try on your own.

Don put flowers in a conical vase for his mother. If the vase has a radius of 4 inches and a height of 15 inches, how much water can it hold?

First of all, what kind of solid figure are we dealing with? The vase is in the shape of a cone. What do we need to find? We need to find the volume of the conical vase in order to know how much water it holds. What information have we been given? We know that the radius of the vase’s base is 4 inches and the height of the vase is 15 inches. Next, we can take the given information and substitute it into the formula for finding the volume of a cone.

VVV=13πr2h=13(3.14)(16)(15)=251.2\begin{align*}V & = \frac{1}{3} \pi r^2 h \\ V & = \frac{1}{3} (3.14)(16)(15) \\ V & = 251.2\end{align*}

The volume of the vase is 251.2 inches3\begin{align*}251.2 \ inches^3\end{align*}.

### Video Review

Here is a video for review.

### Practice

Directions: Find the volume of each cone.

1. A cone with a radius of 3 inches and a height of 7 inches.

2. A cone with a radius of 5 ft and a height of 9 ft.

3. A cone with a radius of 6 meters and a height of 10 meters.

4. A cone with a radius of 10 inches and a height of 12 inches.

5. A cone with a radius of 12 mm and a height of 14 mm.

6. A cone with a radius of 5 ft and a height of 12 feet.

7. A cone with a radius of 4.5 inches and a height of 7 inches.

8. A cone with a radius of 3.5 inches and a height of 5.5 inches.

9. A cone with a radius of 7 cm and a height of 13 cm.

10. A cone with a radius of 8 cm and a height of 11 cm.

11. A cone with a radius of 7.5 inches and a height of 11 inches.

12. A cone with a radius of 11.5 inches and a height of 15 inches.

13. A cone with a radius of 12.5 feet and a height of 15 feet.

Directions: Solve each word problem.

14. A cone has a radius of 6 meters and a height of 14 meters. What is its volume?

15. The containers of icing for Tina’s cake decorator are cones. Each container has a radius of 2.4 inches and a height of 7 inches. If Tina buys containers of red, yellow, and blue icing, how much icing will she buy?

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