# Volume of a Pyramid

This article presents an intuitive method for finding the equation for the volume of a pyramid. It uses an intuitive grasp of the concept of limits and is meant as a potential first encounter with the concepts of calculus.

### Intended Audience

This lesson is for kids. Please imagine yourself as a child learning this for the first time. Would this have been useful to you when you were learning years ago? Would you have benefited from encountering limits this early, even if in an imprecise manner? What about your peers, who perhaps struggled more?

## Introduction

### Finding the Volume of a Pyramid: Your First Encounter

To start, I'd like to invite you to try to come up with a solution on your own. Take a few minutes and think about how you have found equations for the volumes of other 3 dimensional shapes (cubes, prisms, etc.). Can you think of any way to reuse or extend one of those techniques? Can you create your own method? Please don't work too hard. If you find yourself getting frustrated, take a break or move on to the next section. There are some nice pointers to get you started. Please stop often and reassess if you have new ideas about how to solve the problem. The fewer hints you use, the stronger your creative and problem-solving skills will become. Remember there are lots of ways to solve these problems. If you invent one for yourself, which is not covered here, that is awesome!

### An Experimental Approach

Some of the relationships can be guessed by filling a small, hollow three-dimensional solid (with liquid, sand, etc.) and seeing how many pours from the smaller three-dimensional solid it takes to fill a larger three-dimensional solid. For instance, it can be instructive to find a pyramid and prism which have the same base and height and see how many full pyramids it takes to fill the prism. It's important to make sure that the shapes are manufactured to hold the correct volumes though. Based on the reviews, I believe this set meets the criteria. Amazon Link to Hollow 3-Dimensional Solids Measuring precisely and having the prism fill exactly makes for a better "ah-ha moment." If you have a higher tolerance for imprecision, other options like DIY/homemade solids can be used too (ie. made from cardboard and filled with sand).

### Shortcuts

Students who are familiar with algebra, deriving equations, and the concept of direct variation, may find many shortcuts throughout this lesson. Students who have met Cavalieri's principle may notice a variation on it being used in this lesson.

## Finding the Volume of A Square Pyramid (With a Set-Height and Peak Position)

Try considering a square pyramid, whose height is equal to the length of one of the sides of the square base. It may also be useful to consider placing the peak of the pyramid directly above one of the corners of the square base. Can you find the volume of this pyramid?

### Setup

Have a look at this app. Leave the hint buttons alone for now. You can use the full-screen button if you like. There's a reset button at the top of the app, in case you need it. Try playing with the red, green, and blue buttons. What do they do? How might this help you discover the volume of the pyramid? If you would like, drag the cube to rotate it. This may give you a better view of the situation.

## Finding the Volume of A Rectangular Pyramid (With Any Height and Any Peak Position)

In the last section, you found the equation for the volume of a very particular pyramid. In this section, you will generalize this result to a pyramid of any height, with any rectangular base, and with its peak in any position.

### Setup

There are three main objectives to achieve (peak placement, height, and the rectangular base). Before opening the app, consider how you might go about addressing each of these. When you are satisfied, have a look at the app.

Feel free to play with the app. Each of the dials serves a different purpose. Can you figure out what the purpose of each dial is? The app can struggle when the red dial is set to a large number (especially on old or handheld devices). Please be patient. If you have performance issues, consider dropping the value of the red dial while adjustments are made, then raise it again after.

It is of particular importance that you notice what happens when the red dial is set to a large number. I will explain it in the next sentence, so please take some time now to ensure that you have made a strong attempt to understand how the figure changes. As the red dial increases, the number of layers of blocks increases, and the figure starts to resemble a pyramid.

The pyramid's peak can also be dragged to other positions.

Now that you've explored the app, take a moment and ponder how you might use it to complete your three main objectives. Each objective has its own set of hints.

Be sure to reset all three white dials to 1 before exploring each objective.

### Peak Position

### Pyramid Height

### Rectangular Base

Now, just like in the subsection about the height of the pyramid, how would you deal with the prism that encloses the pyramid?

## Finding the Volume of A Regular N-gonal Pyramid (and Stretching to Make Some Non-regular N-gons)

Can you think of a way to find the volume of a regular n-gonal pyramid without looking at the app?

### Setup

Have a look at the app. Feel free to play with the dials. What do you see? The peak is draggable in this app too.

Can you think of a way to find the equation for the volume of a regular n-gonal pyramid given this setup?

You can vary the position of the peak and stretch the pyramid in much the same way as in the previous section. The arguments for why this is valid are very similar. You can even approximate the n-gonal pyramids and prisms with little blocks and stretch those little blocks the way you did in the previous section.

### Alternate Approach

You could also have imagined a square prism enclosing the square pyramid (much like you did in the previous section). You could have then applied the ratio to this square prism to get the n-gonal prism that encloses the n-gonal pyramid you are interested in. Because the ratio is applied to both the prism and the pyramid, it does not change the relationship between the volume of the prism and the pyramid. So you still get this equation.

Using this equation you can then derive , by inserting the the equation for the volume of the prism (ie. by replacing the volume of the prism with area times height).

## A Different Approach

Have a look at the app and play around with the settings. The peak is draggable again. How might this be used as a starting point for finding all of the equations again? Try setting the dials to 30, 0.5, 0.5, 0.5, and 5. You should have 30 layers of the smallest pentagonal slices possible. Move the top layers so that you can see the bottom layers clearly. This may take a lot of dragging. You may also want to look at it using fullscreen. When you look at these bottom layers, do you see anything interesting?

If you look closely, these bottom layers appear to be becoming pentagonal prisms. The pentagonal faces on the top and bottom do not differ much. This is because the slope of the pyramid has a very short distance in which to change them. This short distance is achieved by splitting the pyramid into many layers.

From here, you can work backward through the ideas you explored in the previous sections, eventually finding the relationship between the cube and the particular pyramid from the beginning of the first section (that three square pyramids can be used to make a cube). You can then proceed forward again through the same arguments presented in the previous sections. They will, of course, be easier though, because you will have already done the heavy lifting when you were going backward.

This approach is more roundabout and inefficient, but perhaps matches the natural course of inquiry that one might take if they did not have a structured lesson to guide them.

## What About Cones?

Look at either of the final two apps and make a conjecture about the equation for the volume of a cone. Why do you believe this conjecture is true? The bottom dial (the red one) may be of particular interest.

## Extending to More Complicated Bases

Now that you have experience with this approach to understanding the volumes of pyramids, try thinking about more complicated shapes. What do you think the equation is for the volume of a pyramid whose base is a scalene triangle? What about other n-gons that have less symmetry? What about a crescent moon? Do the same methods still work? Why do you believe this?

Could you apply the same methods to a squiggly closed curve drawn in the following manner? Take a closed loop of string and arrange it on a piece of paper in any manner you want, except the string must not touch itself. This includes not twisting the loop (ie. the string must lay on the paper entirely, it cannot cross over itself). Draw this loop. If this loop were the base of a pyramid could you find an equation for its volume?

## For Teachers and Parents

### What is this Hint System

The hint system is designed to allow students to engage with the problems with little to no support (Discovery/Inquiry-Based Math). The hints gradually give more and more information until we end up with a full explanation (Traditional Lecture). Since Math is not a "spectator sport," it is hoped that repeated use of this hint system, in multiple lessons, will encourage students to use fewer and fewer hints and learn to solve problems creatively for themselves. This is, of course, only one possible arrangement of these hints. An instructor will be better able to adapt the presentation of the hints to progress a specific student has made. This is beneficial to the student, until they learn to use the system and self regulate. If the student needs hints, they can choose to set aside their particular approach temporarily while filling in the gaps in the order the text presents them. Later they can analyze how the approach they took compares to that of the text.

### (Infinitely) Thin Layers? What is this?

I'm blurring the lines between the concept of limit in calculus and Cavalieri's principle. They are already very closely related. The idea is to get the kids thinking about limits in an intuitive fashion before they see them formally in a calculus class. It requires me to be careful in how I present concepts though, as intuition about infinity is very slippery. The details of where things can go wrong have been swept under the rug until a future calculus class, should the student ever take one.

### Why no Rigor?

Sorry, the treatment is not rigorous. To my Pure Math friends, sometimes rigor can get in the way of clarity when first encountering a concept. I promise a rigorous treatment of calculus, straight "outta" Real Analysis, will come in future years.