10.8: Volume of Pyramids and Cones
Introduction
Questions about Ice Cream Cones
Candice and Trevor have been working together for two weeks and 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 could 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 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 \begin{align*}= H = 3.25" \ D = 2.5"\end{align*}
Sugar Cone \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 lesson. By the end of it, you will know which cone holds the most ice cream.
What You Will Learn
In this lesson, you will learn how to do the following:
- Recognize the relationship between the volume of a pyramid and the volume of a prism with the same base and height.
- Find volumes of pyramids using formulas.
- Recognize the relationship between the volume of a cone and the volume of a cylinder with the same base and height.
- Find volumes of cones using formulas.
- Solve real-world problems involving volumes of cones and pyramids.
Teaching Time
I. Recognize the Relationship between the Volume of a Pyramid and the Volume of a Prism with the Same Base and Height
In this lesson we will learn to find the volume of pyramids and cones. Recall that pyramids and cones are solid shapes that exist in three-dimensional space. A pyramid has sides that are triangular faces and a polygon base. The base can be any shape. Let’s look at some pyramids.
Like pyramids, cones have a base and a point at the top. However, cones always have a circular base. They have only one side, and it is curved. Here is a picture of a cone.
Volume is the measure of how much space a three-dimensional figure takes up or holds. Imagine a funnel. Its size determines how much water the funnel will hold. If we fill it with water, the amount of water represents the volume of the funnel. We measure volume in cubic units, because we are dealing with three dimensions: length, width, and height.
Let’s start by looking at how we can find the volume of a pyramid.
We can start by comparing a pyramid with a figure that is similar to it. We know that a prism has a length, a width and a height. The parallel bases of a prism can be any polygon, this one has a square base, so we call it a cube.
Now think about a pyramid. It has a base that can be any polygon and the base of this pyramid is a square.
Wow! These are both similar given that they have square bases. It may help your understanding of the volume of a pyramid to compare it to the volume of the cube or prism. Here is a picture of it.
If we were going to find the volume of this cube, we would multiply the area of the base by the height. To find the volume of the pyramid, we take the area of the base, \begin{align*}B\end{align*} and multiply it by the height and then multiply it by \begin{align*}\frac{1}{3}\end{align*}. Here is the formula for finding the volume of a pyramid.
\begin{align*}V=\frac{1}{3} Bh \end{align*}
Yes it does. Thinking about the volume of the pyramid in this way will help you to understand the formula and why it works.
Take a few minutes and copy this formula down in your notebook. Make sure to note that this is the formula for finding the volume of a pyramid.
II. Find Volumes of Pyramids using Formulas
Now that we have the formula and some understanding about the volume of a pyramid, let’s put what we have learned to work. We can use the information that we have gathered so far to find the volume of different pyramids.
One thing to keep in mind is that pyramids can be tricky because they can have many different bases. Look at the formula again.
\begin{align*}V=\frac{1}{3} Bh \end{align*}
That capital \begin{align*}B\end{align*} means that we need to find the area of the base. If the base is a square, we will need to use one formula. If the base is a triangle, we would need to use a different formula and so on. Making sure that we have the correct formula is essential in our work with pyramids.
Let’s look at an example.
Example
What is the volume of the pyramid below?
First, let’s decide what shape the base of the pyramid is. One side is 10 centimeters and the other is 6 centimeters, so it must be a rectangle. We need to use the area formula for rectangles to find \begin{align*}B\end{align*}, the base area.
\begin{align*}B & = lw\\ B & = 10 (6)\\ B & = 60 \ cm^2\end{align*}
The area of this pyramid’s base is 60 square centimeters. Now we multiply this by the height and \begin{align*}\frac{1}{3}\end{align*}, according to the formula.
\begin{align*}V & = \frac{1}{3} Bh\\ V & = \frac{1}{3} (60) (15)\\ V & = 20 (15)\\ V & = 300 \ cm^3\end{align*}
The volume of the pyramid is 300 cubic centimeters.
Remember, we measure volume in three dimensions, so we write the answer in cubic units.
Example
Find the volume of the figure below.
What is the shape of the base? This time it is a triangle, so we’ll need to use the area formula for triangles to find the base area. Be careful not to confuse the height of the base triangle with the height of the whole pyramid!
\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (8) (3)\\ B & = 4 (3)\\ B & = 12 \ in.^2 \end{align*}
The base area for this triangular pyramid is 12 square inches. Let’s put this into the formula and solve for \begin{align*}V\end{align*}, volume.
\begin{align*}V & = \frac{1}{3} Bh\\ V & = \frac{1}{3} (12) (17)\\ V & = 4 (17)\\ V & = 68 \ in.^3\end{align*}
The volume of this pyramid is 68 cubic inches.
10 M. Lesson Exercises
Find the volume of the following pyramids. You may round to the nearest hundredth when necessary.
- A square pyramid with a base of 8 cm and a height of 5 cm.
- A rectangular pyramid with a length of 10 cm, a width of 8 cm and a height of 9 cm.
- A square pyramid with a base of 5.5 in and a height of 4 in.
Take a few minutes and check your work with a friend. Did you round when necessary?
III. Recognize the Relationship between the Volume of a Cone and the Volume of a Cylinder with the same Base and Height
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 certainly figures that we can compare.
Here is a cylinder and the formula for finding the volume of a cylinder.
\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.
\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 base of the cone. The formula for this is \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 only fills a portion of the cylinder, we take one-third of the product.
Take a few minutes and copy this formula down in your notebook.
IV. Find Volumes of Cones using Formulas
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. Let’s look at an example.
Example
Find the volume of the following cone.
We can start by substituting the given values into the formula.
\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.
Example
What is the volume of the cone below?
First, we can substitute the given values into the formula.
\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 \begin{align*}113.04 \ cm^3\end{align*}.
10N. Lesson Exercises
Find the volume of each cone using the given values and the formula.
- A cone with a radius of 2 inches and a height of 4 inches.
- A cone with a radius of 5 ft and a height of 8 ft.
- A cone with a radius of 4 m and a height of 9 m.
Take a few minutes to go over your answers with a partner. Correct any errors and then continue with the lesson.
V. Solve Real-World Problems Involving Volumes of Pyramids and Cones
We can also use the formula to solve real-world problems involving volume of pyramids or cones. Be sure you understand what the problem is asking. Then look to see what information is given in the problem. Third, put this information in for the appropriate variable in the formula and solve. Remember always to look first to see what shape the base of the figure is. If a picture is not given, draw one to help you. Let’s give it a try.
Example
Felice bought the candle below for her friend’s birthday. The package says that the candle burns one hour for every 20 cubic centimeters of wax. How many hours will it take the entire candle to burn?
First, let’s determine what the problem is asking us to find. We need to find the number of hours the candle will burn. This depends on how big the candle is, so first we need to find its volume. The volume of the candle is the amount of wax it holds. What information have we been given? We know the dimensions of the base, which is a square, so let’s use the area formula for squares to find the base area.
\begin{align*}B & = s^2\\ B & = (12)^2\\ B & = 144 \ cm^2\end{align*}
The base area of the pyramid is 144 square centimeters. We can put this information into the formula and solve for \begin{align*}V\end{align*}, volume.
\begin{align*}V & = \frac{1}{3} Bh\\ V & = \frac{1}{3} (144) (24)\\ V & = 48 (24) \\ V & = 1,152 \ cm^3\end{align*}
Now we know that the candle contains 1,152 cubic centimeters of wax. But we’re not done yet! Remember, we need to find how many hours the candle will burn. Look back at the problem. It tells us that the candle burns one hour for every 20 cubic centimeters of wax. To find how many hours the candle will burn, we need to divide the total volume of wax by 20.
\begin{align*}1,152 \div 20 = 57.6\end{align*}
The candle will burn for 57.6 hours.
Example
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.
\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 \begin{align*}251.2 \ inches^3\end{align*}.
Real–Life Example Completed
Questions about Ice Cream Cones
Here is the original problem once again. Reread it and underline any of the important information.
Candice and Trevor have been working together for two weeks and 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 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 \begin{align*}= H = 3.25" \ D = 2.5"\end{align*}
Sugar Cone \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.
\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.
\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 found in this lesson.
- 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.
Technology Integration
James Sousa, Determine the Volume of a Cone
Other Videos:
- http://www.mathplayground.com/mv_volume_cones.html – This is a Brightstorm video on how to find the volume of a cone.
- http://www.mathplayground.com/mv_volume_pyramids.html – This is a Brightstorm video on how to find the volume of a pyramid.
Time to Practice
Directions: Find the volume of each of the following pyramids.
1. A square pyramid with a base of 6 ft and a height of 9 ft.
2. A square pyramid with a base of 8 m and a height of 10 m.
3. A square pyramid with a base of 11 in and a height of 13 in.
4. A square pyramid with a base of 9 ft and a height of 14 ft.
5. A square pyramid with a base of 4.5 in and a height of 5 inches.
6. A rectangular pyramid with a base length of 4 in, a base width of 3 in and a height of 5 in.
7. A rectangular pyramid with a base length of 5 ft, a base width of 4 ft and a height of 6 ft.
8. A rectangular pyramid with a base length of 7 m, a base width of 4 m and a height of 9 m.
9. A triangular pyramid with a base length of 5 in and a base height of 4 inches with a pyramid height of 6 inches.
10. A triangular pyramid with a base length of 8 ft and a base height of 7 ft with a pyramid height of 9 ft.
Directions: Find the volume of each cone.
11. A cone with a radius of 3 inches and a height of 7 inches.
12. A cone with a radius of 5 ft and a height of 9 ft.
13. A cone with a radius of 6 meters and a height of 10 meters.
14. A cone with a radius of 10 inches and a height of 12 inches.
15. A cone with a radius of 12 mm and a height of 14 mm.
16. A cone with a radius of 5 ft and a height of 12 feet.
17. A cone with a radius of 4.5 inches and a height of 7 inches.
18. A cone with a radius of 3.5 inches and a height of 5.5 inches.
19. A cone with a radius of 7 cm and a height of 13 cm.
20. A cone with a radius of 8 cm and a height of 11 cm.
Directions: Solve each word problem.
21. A cone has a radius of 6 meters and a height of 14 meters. What is its volume?
22. A square pyramid has a base with sides of 5 yards each and a height of 21 yards. What is its volume?
23. 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?
24. Claire has a perfume bottle shaped like a triangular pyramid. Its base area is 48 square centimeters, and its height is 28 centimeters. How much does the bottle hold when it is exactly half full?
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