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Volume of Pyramids

V = Bh/3 where B is the area of the base

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Volume of Pyramids
License: CC BY-NC 3.0

Brianna bought a unique candle as a gift for her friend’s birthday. The candle has a square base with a length of \begin{align*}12 \ cm\end{align*}. The height of the candle is \begin{align*}24 \ cm\end{align*}. The package says that the candle burns one hour for every 20 cubic centimeters of wax. How many hours will it take for the entire candle to burn?

In this concept, you will learn to calculate the volume of pyramids.

Volume

Volume is the measure of how much space a three-dimensional figure takes up or holds. Imagine a funnel. The funnel’s size will determine how much water the funnel will hold. If you fill it with water, the amount of water tells the volume of the funnel. Volume is often what you think of when you talk about measuring liquid or liquid capacity. Note that you measure volume in cubic units.

Pyramids are unusual because they are so much smaller at the top than they are at their base. The important thing to remember is that measuring volume involves filling up a solid figure. A pyramid has exactly one-third the volume of a cube.

The formula for finding the volume of pyramids is:

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

\begin{align*}B\end{align*} represents the base of the pyramid and \begin{align*}h\end{align*} is the height of the pyramid. Pyramids can have bases of any shape. Pyramids can have triangular, rectangular, or square bases. That means you need to choose the appropriate formula for finding the area of the base, or \begin{align*}B\end{align*}.

Here are some common area formulas:

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

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

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

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

Let’s look at an example.

What is the volume of the pyramid below?

License: CC BY-NC 3.0

First, figure out the shape of the base. There are two pairs of parallel sides that meet at right angles, so it must be a rectangle. You need to use the area formula for rectangles to find \begin{align*}B\end{align*}, the base area.

\begin{align*}\begin{array}{rcl} A &=& lw \\ A &=& (11)(6.3) \\ A &=& 69.3 \end{array}\end{align*}

Next, substitute the base area, \begin{align*}A\end{align*}, and the height into the formula for the volume.

\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (69.3)(15) \\ V &=& 346.5 \end{array}\end{align*}

The answer is 346.5.

The volume of the pyramid is \begin{align*}346.5 \ in^3\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Brianna’s melting gift.

Brianna bought a candle with a square base length of \begin{align*}12 \ cm\end{align*} and a height of \begin{align*}24 \ cm\end{align*}. She wants to know how long the candle will burn at a rate of \begin{align*}20 \ cm^3\end{align*} per hour.

First, use the area formula for squares to find \begin{align*}B\end{align*}, the base area.

\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (12)^2 \\ A &=& 144 \end{array}\end{align*}

Next, substitute the base area, \begin{align*}A\end{align*}, and the height into the formula for the volume.

\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (144)(24) \\ V &=& 1152 \end{array}\end{align*}

Then, find the number of hours the candle will burn.

\begin{align*}\begin{array}{rcl} \# \ \text{hours} &=& \frac{1152}{20} \\ \# \ \text{hours} &=& 57.6 \end{array}\end{align*}

The answer is 57.6.

The candle will burn 57.6 hours.

Example 2

A triangular pyramid has a volume of \begin{align*}266 \ \text{cubic feet}\end{align*} and a base area of \begin{align*}42 \ \text{square feet}\end{align*}. What is its height?

First, substitute what you know into the formula for the volume of the pyramid.

\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ 266 &=& \frac{1}{3}(42)h \end{array}\end{align*}

Next, use algebra to solve for the height, \begin{align*}h\end{align*}.

\begin{align*}\begin{array}{rcl} 266 &=& \frac{1}{3}(42)h \\ 266 &=& 14h \\ \frac{266}{14} &=& \frac{14h}{14} \\ h &=& 19 \end{array}\end{align*}

The answer is 19.

The height of the pyramid is \begin{align*}19 ft\end{align*}.

Example 3

Find the volume of a square pyramid with a base of \begin{align*}5.5 \ in\end{align*} and a height of \begin{align*}4 \ in\end{align*}.

First, use the area formula for squares to find \begin{align*}B\end{align*}, the base area.

\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (5.5)^2 \\ A &=& 30.25 \end{array}\end{align*}

Next, substitute the base area, \begin{align*} A\end{align*}, and the height into the formula for the volume.

\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (30.25)(4) \\ V &=& 40.33 \end{array}\end{align*}

The answer is 40.33.

The volume of the pyramid is \begin{align*}40.33 \ in^3\end{align*}.

Example 4

Find the volume of a square pyramid with a base of \begin{align*}8 \ cm\end{align*} and a height of \begin{align*}6 \ cm\end{align*}.

First, use the area formula for squares to find \begin{align*}B\end{align*}, the base area.

\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (8)2 \\ A &=& 64 \end{array}\end{align*}

Next, substitute the base area, \begin{align*}A\end{align*}, and the height into the formula for the volume.

\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (64)(6) \\ V &=& 128 \end{array}\end{align*}

The answer is 128.

The volume of the pyramid is \begin{align*}128 \ cm^3\end{align*}.

Example 5

Find the volume of a rectangular pyramid with a length of \begin{align*}10 \ cm\end{align*}, a width of \begin{align*}8 \ cm\end{align*} and a height of \begin{align*}9 \ cm\end{align*}.

First, use the area formula for rectangles to find \begin{align*}B\end{align*}, the base area.

\begin{align*}\begin{array}{rcl} A &=& lw \\ A &=& (10)(8) \\ A &=& 80 \end{array}\end{align*}

Next, substitute the base area, \begin{align*}A\end{align*}, and the height into the formula for the volume.

\begin{align*}\begin{array}{rcl} V &=& \frac{1}{3} Bh \\ V &=& \frac{1}{3} (80)(9) \\ V &=& 240 \end{array}\end{align*}

The answer is 240.

The volume of the pyramid is \begin{align*}240 \ cm^3\end{align*}.

Review

Answer each of the following questions.

1. What is the formula that you would need for finding the volume of a pyramid?

2. When you see a capital B in a formula it means that you are looking for the perimeter or area of the base?

3. What is the name of the figure pictured below?

4. What is the shape of the base?

5. What is the volume of the figure?

License: CC BY-NC 3.0

License: CC BY-NC 3.0

6. What is the name of the figure pictured above?

7. What is the difference between this figure and the last figure?

8. What is the volume of this figure?

9. What is the shape of the base?

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

10. A square pyramid has a base with sides of 6 yards each and a volume of 175 cubic yards. What is its height?

11. 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?

12. Find the volume of a square pyramid with a base of 4 inches and a height of 6 inches.

13. Find the volume of a rectangular pyramid with a length of 5 inches, a width of 7 inches and a height of 8 inches.

14. Find the volume of a square pyramid with a base of 8 meters and a height of 12 meters.

15. Find the volume of a square pyramid with a base of 13.5 meters and a height of 15 meters.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.12. 

Resources

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Vocabulary

Pyramid

A pyramid is a three-dimensional object with a base that is a polygon and triangular faces that meet at one vertex.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

Image Attributions

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