# 10.14: Volume of Pyramids

Difficulty Level: At Grade Created by: CK-12
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Have you ever bought perfume in a bottle?

Claire has purchased 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?

To figure this out, you will need to know how to find the volume of a pyramid. Pay attention and you will know how to do this by the end of the Concept.

### Guidance

In this Concept, we will learn to find the volume of pyramids. Pyramids 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.

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 tells 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 we can 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. We can think about the volume of a pyramid by looking at the volume of the cube or prism. Let’s look at this picture of it.

If we were going to find the volume of this cube, we would multiply the base times the width times the height. To find the volume of the pyramid, we take the area of the base, \begin{align*}B\end{align*} and multiply it times 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 notebooks. Make sure to note that this is the formula for finding the volume of a pyramid.

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 applying this information.

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 with \begin{align*}a^3\end{align*}.

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.

Find the volume of the following pyramids. You may round to the nearest hundredth when necessary.

#### Example A

A square pyramid with a base of 8 cm and a height of 6 cm.

Solution: \begin{align*}128 \ cm^3\end{align*}

#### Example B

A rectangular pyramid with a length of 10 cm, a width of 8 cm and a height of 9 cm.

Solution: \begin{align*}240 \ cm^3\end{align*}

#### Example C

A square pyramid with a base of 5.5 in and a height of 4 in.

Solution: \begin{align*}40.33 \ in^3\end{align*}

Here is the original problem once again.

Claire has purchased 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?

First, notice that we have been given the area of the base of the perfume bottle. We don't have to figure that out. That is the capital B in our formula.

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

Now let's solve for the full volume of the bottle.

\begin{align*}V = \frac{1}{3}(48)(28)\end{align*}

\begin{align*}V = \frac{1}{3}(1344)\end{align*}

\begin{align*}V = 448 \ cm^3\end{align*}

This is the volume of a full bottle. However, the problem tells us that we need to figure out the volume if the bottle is half - full. We can divide our volume by 2.

\begin{align*}V = 448 \div 2 = 224\end{align*}

The volume of a half - full bottle of perfume is \begin{align*}224 \ cm^3\end{align*}.

### 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

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.

### Video Review

Here is a video for review.

### 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.

11. A square pyramid with a base of 8 feet and height of 4 feet.

12. A rectangular pyramid with a length of 5 inches, a width of 4 inches and a height of 6 inches.

13. A square pyramid with a base of 3.5 feet and a height of 6.5 feet.

14. A square pyramid with a base of 6.5 feet and a height of 8.5 feet.

15. A rectangular pyramid with a width of 4 feet, a length of 6 feet and a height of 7.5 feet.

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