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8.7: Volume of Pyramids and Cones

Difficulty Level: At Grade Created by: CK-12

Introduction

Ice Cream Sales

“We need a fundraiser,” Maria said at the planning meeting for the Olympics.

“I agree, and besides, people like to eat,” Jamie agreed.

“How about ice cream cones? We can use the freezer in the lunch room and scoop and serve,” Dan suggested.

“I think that’s a great idea. How about using waffle cones?” Maria added.

The group continued to discuss the ice cream cones and finally agreed on two different sized cones, one that is 4\begin{align*}4^{\prime\prime}\end{align*} in diameter and 4\begin{align*}4^{\prime\prime}\end{align*} long and one that is 5\begin{align*}5^{\prime\prime}\end{align*} in diameter and 6\begin{align*}6^{\prime\prime}\end{align*} long.

“We can charge double for the larger cone,” Jamie said.

“I don’t think so. It isn’t double the size,” Dan disagreed.

“But it will hold double the amount of ice cream,” Jamie explained.

“I don’t think so because it isn’t twice as large.”

“That doesn’t matter when it comes to volume,” Jamie said.

Who is correct? To figure this out, you will need to find the volume of both cones. Then you will be able to decide whether the group can charge double for the larger cone.

What You Will Learn

By the end of this lesson, you will be able to complete the following skills.

• Recognize volume of pyramids and cones as the sum of volumes of layers of unit cubes.
• Find volumes and linear dimensions of pyramids with any polygonal base.
• Find volumes and linear dimensions of cones and truncated cones.
• Solve real – world problems involving volume of pyramids and cones, including metric and customary units of capacity.

Teaching Time

I. Recognize Volume of Pyramids and Cones as the Sum of Volumes of Layers of Unit Cubes

In this lesson we will learn to find the volume of pyramids and cones. Pyramids and cones are solid shapes that exist in three-dimensional space. A pyramid has sides that are triangular faces and a base. The base can be any shape. 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.

What is volume?

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 tells the volume of the funnel. Volume is often what we think of when we talk about measuring liquid or liquid capacity.

In the last lesson, you learned how to calculate the volume of prisms and cylinders. In this lesson, you will learn how to calculate the volume of pyramids and cones.

We measure volume in three dimensions: length, width, and height. We therefore measure volume in cubic units. We can use unit cubes to represent volume. Have a look at the cube below.

You can see that the cube is 3×3×3\begin{align*}3 \times 3 \times 3\end{align*}. If we wanted to find the volume of this cube, we could find the area of the base and then multiply it by the height.

VVVV=Bh=(s2)3=32(3)=27 cubic units\begin{align*}V & =Bh\\ V & =(s^2 )3\\ V & =3^2 (3)\\ V & =27 \ cubic \ units\end{align*}

Notice that we measure volume in cubic units.

Pyramids and cones are unusual, however, because they are so much smaller at the top than they are at their base. It becomes very difficult to use unit cubes to measure the volume of these solids because we would be calculating parts of unit cubes.

The important thing to remember is that measuring volume involves filling up a solid figure.

The pyramid and cube have bases with the same area, as do the cone and the cylinder. But what happens when we try to add the third dimension of height? In the cube and cylinder, we “stack” the base, as if in layers. In the pyramid and cone, however, the area gets smaller as we go up. In fact, it gets smaller in equal proportion as we add each layer. A pyramid has exactly one-third the volume of a cube. A cone has exactly one-third the volume of a cylinder.

Notice that there is a relationship between the pyramid and the cube and the cylinder and the cone. This is where we get the formula for finding the volume of both of these solids. Here is the formula for finding the volume of pyramids and cones.

V=13Bh\begin{align*}V= \frac{1}{3} Bh\end{align*}

Let’s look at how to use this formula to find the volume of pyramids first.

II. Find Volumes and Linear Dimensions of Pyramids with Any Polygonal Base

As we have seen, we are dealing with three dimensions when we find the volume of a solid figure. Finding the area of the base accounts for two of the dimensions the length and the width. Then we multiply this by the height of the figure. Because a pyramid has 13\begin{align*}\frac{1}{3}\end{align*} the volume of a cube, we use the cube formula and then multiply by 13\begin{align*}\frac{1}{3}\end{align*}. We write the volume formula like this:

V=13Bh\begin{align*}V = \frac{1}{3} Bh\end{align*}

In the formula, V\begin{align*}V\end{align*} stands for volume, the amount we are solving for. B\begin{align*}B\end{align*} represents the base area, and h\begin{align*}h\end{align*} represents the height of the pyramid.

Pyramids can be tricky, however, because they can have bases of any shape. Pyramids can have triangular, rectangular, or square bases. That means we need to choose the appropriate formula for finding the area of the base, or B\begin{align*}B\end{align*}. Here are the common area formulas:

Square: A=s2\begin{align*}A = s^2\end{align*}

Rectangle: A=lw\begin{align*}A = lw\end{align*}

Triangle: A=12bh\begin{align*}A = \frac{1}{2} bh\end{align*}

When given a pyramid, the first thing we need to do is determine the shape of the base. Then we’ll know which formula to use to find the base area. Once we have the base area, we put it into the volume formula along with the height of the pyramid and then solve for V\begin{align*}V\end{align*}. Let’s give it a try.

Example

What is the volume of the pyramid below?

First, let’s decide what shape the base of the pyramid is. There are two pairs of parallel sides that meet at right angles, so it must be a rectangle. We need to use the area formula for rectangles to find B\begin{align*}B\end{align*}, the base area.

BBB=lw=11(6.3)=69.3 cm2\begin{align*}B & = lw\\ B & = 11 (6.3)\\ B & = 69.3 \ cm^2\end{align*}

The area of this pyramid’s base is 69.3 square centimeters. Now we multiply this by the height and 13\begin{align*}\frac{1}{3}\end{align*}, according to the formula.

VVVV=13Bh=13(69.3)(15)=23.1(15)=346.5 cm3\begin{align*}V & = \frac{1}{3} Bh\\ V & = \frac{1}{3} (69.3) (15)\\ V & = 23.1 (15)\\ V & = 346.5 \ cm^3\end{align*}

The volume of this pyramid is 346.5 cm3\begin{align*}346.5 \ cm^3\end{align*}. Remember that volume is always measured in cubic units and that is why our exponent is a three.

We can also work the other way around. If you know the base area and the volume, then you can use substitution and our formula to find the missing measurement for height. Let’s look at an example.

Example

A triangular pyramid has a volume of 266 cubic feet and a base area of 42 square feet. What is its height?

What do we need to find? We need to solve for the height, h\begin{align*}h\end{align*}. We have been given the volume and the base area so we simply put this information into the formula.

V266266266÷1419=13Bh=13(42)h=14h=h=h\begin{align*}V & = \frac{1}{3} Bh\\ 266 & = \frac{1}{3} (42)h\\ 266 & = 14h\\ 266 \div 14 & = h\\ 19 & = h\end{align*}

The height of this pyramid is 19 feet.

Now let’s look at how we can use a formula to find the volume of cones.

III. Find Volumes and Linear Dimensions of Cones and Truncated Cones

We can use the same general formula to find the volume of cones: V=13Bh\begin{align*}V = \frac{1}{3} Bh\end{align*}. However, cones have a circular base. To find the base area, then, we need to use the formula for finding the area of a circle: A=πr2\begin{align*}A = \pi r^2\end{align*}. When working with pyramids, the polygon base could be different depending on the type of pyramid. This is different from a cone. With a cone, you will always have a circular base, so you will always be using the same formula for area to find the base. Here is what it would look like as one formula.

V=13(πr2)(h)\begin{align*}V=\frac{1}{3} (\pi r^2)(h)\end{align*}

Now you can see that we would find the area of the base, multiply it by the height and then multiply it by one-third or take one-third of the product of the base area and the height.

Let’s apply this with an example.

Example

What is the volume of the cone below?

First, we need to find the base area. The base is a circle, so we use the area formula for circles.

BBBB=πr2=π(3.52)=12.25π=38.47 cm2\begin{align*}B & = \pi r^2\\ B & = \pi (3.5^2)\\ B & = 12.25 \pi\\ B & = 38.47 \ cm^2\end{align*}

The circular base has an area of 38.47 square centimeters. Now we can put this measurement into the formula for volume.

VVVV=13Bh=13(38.47)(22)=12.82(22)=282.04 cm3\begin{align*}V & = \frac{1}{3} Bh\\ V & = \frac{1}{3} (38.47) (22)\\ V & = 12.82 (22)\\ V & = 282.04 \ cm^3\end{align*}

The volume of this cone is 282.04 cm3\begin{align*}282.04 \ cm^3\end{align*}.

We can find a missing dimension of a cone if we have been given the volume of that cone and the base area. Let’s look at an example.

Example

What is the height of a cone whose radius is 1.6 meters and volume is 20.1 cubic meters?

What information have we been given, and what do we need to find? We know the radius, so we can calculate the base area. We also know the volume, so we can put this into the formula and solve for h\begin{align*}h\end{align*}, the height. Let’s find B\begin{align*}B\end{align*} first.

BBBB=πr2=π(1.62)=2.56π=8.04 m2\begin{align*}B & = \pi r^2\\ B & = \pi (1.6^2)\\ B & = 2.56 \pi\\ B & = 8.04 \ m^2\end{align*}

The base area is 8.04 square inches when we approximate pi. Now let’s put this into the volume formula.

V20.120.120.1÷2.687.5 m=13Bh=13(8.04)h=2.68h=h=h\begin{align*}V & = \frac{1}{3} Bh\\ 20.1 & = \frac{1}{3} (8.04)h\\ 20.1 & = 2.68h\\ 20.1 \div 2.68 & = h\\ 7.5 \ m & = h\end{align*}

We found that the height of the cone must be 7.5 meters.

In other sections on figuring out volume, we have also worked with pieces of other solids. These figures are referred to as truncated solids. We can find the volume of a truncated cone as well. Let’s explore this a little further.

That is a great question. Take a look at this image to understand what a truncated cone looks like.

Notice that we have two radii to work with and the height of the truncated cone as well. We can use the following formula to calculate volume.

V=13π(r12+(r1)(r2)+r22)h\begin{align*}V=\frac{1}{3} \pi(r1^2+(r1)(r2)+r2^2)h\end{align*}

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

Now we can take some measurements and figure out the volume of the truncated cone. Let’s look at an example.

Example

What is the volume of a truncated cone with a top radius of 2 cm, a bottom radius of 4 cm and a height of 4.5 cm?

To work through this problem, we have to substitute the given values into the formula and solve for the volume.

VVVVVV=13π(r12+(r1)(r2)+r22)h=13π(22+(2)(4)+42)4.5=13π(4+8+16)4.5=13π(126)=13(395.64)=131.88 cm3\begin{align*}V & = \frac{1}{3} \pi (r1^2+(r1)(r2)+r2^2)h \\ V & = \frac{1}{3} \pi (2^2+(2)(4)+4^2)4.5 \\ V & = \frac{1}{3} \pi (4+8+16)4.5 \\ V & = \frac{1}{3} \pi (126)\\ V & = \frac{1}{3} (395.64)\\ V & = 131.88 \ cm^3 \end{align*}

This is the volume of this truncated cone.

IV. Solve Real – World Problems Involving Volume of Pyramids and Cones, Including Metric and Customary Units of Capacity

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

Brianna 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. For how many hours will it take for 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. This is our answer.

Now let’s go back and solve the problem from the introduction.

Real-Life Example Completed

Ice Cream Sales

Here is the original problem from the introduction. Reread it and then figure out the volume of both ice cream cones. Finally decide if the group can charge double for the larger cone-be sure to justify your answer.

“We need a fundraiser,” Maria said at the planning meeting for the Olympics.

“I agree, and besides, people like to eat,” Jamie agreed.

“How about ice cream cones? We can use the freezer in the lunch room and scoop and serve,” Dan suggested.

“I think that’s a great idea. How about using waffle cones?” Maria added.

The group continued to discuss the ice cream cones and finally agreed on two different sized cones, one that is \begin{align*}4^{\prime\prime}\end{align*} in diameter and \begin{align*}4^{\prime\prime}\end{align*} long and one that is \begin{align*}5^{\prime\prime}\end{align*} in diameter and \begin{align*}6^{\prime\prime}\end{align*} long.

“We can charge double for the larger cone,” Jamie said.

“I don’t think so. It isn’t double the size,” Dan disagreed.

“But it will hold double the amount of ice cream,” Jamie explained.

“I don’t think so because it isn’t twice as large.”

“That doesn’t matter when it comes to volume,” Jamie said.

Solution to Real – Life Example

We can begin by figuring out the volume of both ice cream cones.

Cone 1 has a diameter of \begin{align*}4^{\prime\prime}\end{align*} and a height of \begin{align*}4^{\prime\prime}\end{align*}

Cone 2 has a diameter of \begin{align*}5^{\prime\prime}\end{align*} and a height of \begin{align*}6^{\prime\prime}\end{align*}

The formula for volume of a cone is \begin{align*}\frac{1}{3} \pi r^2 h\end{align*}

Cone 1

\begin{align*}\frac{1}{3}(3.14)(2^2)(4) = 16.75 \ in^3\end{align*}

Cone 2

\begin{align*}\frac{1}{3}(3.14)(2.5^2)(6) = 39.25 \ in^3\end{align*}

The volume of Cone 2 is more than double that of Cone 1. The students could definitely charge double for the cone if they chose to.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Volume
the capacity inside a solid figure or the amount of space a solid figure can hold.
Pyramid
a solid figure with a polygon base and triangular side faces that meet in a single vertex.
Cones
a circular base and a curved side that meets in a single vertex.
Base Area
the area of the base of a solid figure.
Height
the measurement that is perpendicular to the base of a solid figure.
Truncated cone
a section of a cone-it has two circular radii – one on top and one as a base.

Time to Practice

Directions: Answer each of the following questions.

1. What is the formula for finding the volume of a cone?
2. True or false. A truncated cone is a cone without a vertex.
3. True or false. You can use the same formula to find the volume of a truncated cone as a regular cone.
4. What is the formula that you would need for finding the volume of a pyramid?
5. When you see a capital B in a formula it means that you are looking for the perimeter or area of the base?

Directions: Look at each figure and then answer the following questions about each.

1. What is the name of the figure pictured above?
2. What is the diameter of the base?
3. What is the volume of this figure?

1. What is the name of the figure pictured above?
2. What is the shape of the base?
3. What is the volume of the figure?

1. What is the name of the figure pictured above?
2. What is the difference between this figure and the last figure?
3. What is the volume of this figure?
4. What is the shape of the base?

1. What is the diameter of this cone?
2. What is the height of the cone?
3. What is the volume of the cone?

Directions: Use what you have learned to solve each of the following problems.

1. A cone has a radius of 6 meters and a volume of \begin{align*}168 \pi\end{align*}. What is its height?
2. A square pyramid has a base with sides of 5 yards each and a volume of 175 cubic yards. What is its height?
3. 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?
4. 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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Jan 14, 2013
Apr 29, 2014
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