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.
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?
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).
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?
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.
Digital Manipulative: Volume of a Particular Square Pyramid
Feel free to use the hint button. Does this new perspective help?
Is there a relationship between the three colored pyramids?
How do their volumes compare? Rotating the cube about its diagonal may help to emphasize certain symmetries.
The pyramids are identical in shape, therefore their volumes are the same. How does this help?
How do the volumes of the three pyramids compare to the volume of the cube?
The cube can be constructed from the three pyramids. This means that the volume of the cube is equal to the volume of all three pyramids combined.
How can the volume of the pyramid be expressed in terms of the volume of the cube?
The volume of the pyramid is one-third the volume of the cube.
Using your previous knowledge about the volumes of cubes, what other ways might you express the volume of the pyramid?
It might be useful to consider different ways to express the volume of a cube.
The volume of a cube can be expressed as the side length to the third power.
So the volume of the pyramid can be expressed as one-third times the side length to the third power.
While true, this particular expression will not be useful later in this lesson. Can you think of another expression for the volume of a cube (and thus the pyramid)?
Another expression for the volume of a cube is length times width times height.
So the volume of the pyramid can be expressed as one-third times the length times the width times height.
Can you think of another pair of expressions?
Another expression for the volume of a cube is the area of the base times the height.
So the volume of the pyramid can be expressed as one-third times the area of the base times the height.
This equation comes from the fact that the cube is a special kind of square prism.
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.
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.
Digital Manipulative: Volume of a Rectangular Pyramid
Reset all three white dials to 1. How does shifting the peak affect the volume? Why?
How does shifting the peak affect the volume of each individual layer of blocks?
The volume of each layer does not change. What does this mean for the overall pyramid?
The layers approximate the pyramid. Since the volume of each layer remains constant, the volume of the overall pyramid remains constant. How can you use this to compare the volume of this pyramid to the volume of the square pyramid from the previous section?
If you first set all three white dials to 1 and place the peak over one of the corners, then this pyramid is now the same as the pyramid from the previous section. We can shift the peak anywhere we want without changing volume. Now, what is the equation for the volume of these pyramids?
The equations are exactly the same as for the previous section.
Reset all three white dials to 1. What does changing the middle white dial do?
Trying values of the red dial which are large and small may help you understand. What do you see when you vary the middle white dial and the red dial is set to a large number? What do you see when you vary the middle white dial and the red dial is set to a small number? The effects are visible in both cases, however, it's easier to see each of the two effects when the number of layers is at extreme values.
When the red dial is set to a large number, it's easier to see the height of the pyramid change when the value of the middle white dial changes. When the red dial is set to a small number it's easier to see the height of the individual layers change when the value of the middle white dial changes. How much change occurs?
When the dial doubles, the height of the pyramid doubles. When the dial halves the height of the pyramid halves. How can you use this to find an equation for the volume of this pyramid?
In the previous section you compared the volume of the pyramid to the volume of a cube, can you think of a way to do something similar here?
In the previous section, the cube had the same dimensions as the pyramid you were interested in. This allowed the pyramid to fit perfectly inside the cube. The pyramid was as large as it could be without extending beyond the cube. Assuming a reasonable peak position, what shape should you put around your new pyramid?
Assuming the top and bottom white dials have been set correctly (to 1), it might be interesting to try a square prism whose base matches the base of the pyramid and whose height matches the height of the pyramid. If you would like to visualize this prism, set the red dial to one.
How does the height of this prism vary as the height of the pyramid varies? Try varying with the second white dial for a hint.
The height of both the pyramid and the prism match at all times. How does the prism's volume vary as compared to the cube that we started with (when all three white dials are set to 1)?
Doubling the height appears to double the volume. Halving the height appears to halve the volume. How does this compare with how the pyramid's volume varied?
In both the case of the pyramid and the prism, it appears doubling the height or halving the height doubles the volume or halves the volume respectively. What does this mean for the equation for the volume of the pyramid?
Consider the small blocks that make up each layer of the pyramid, and imagine similar blocks making up layers of the prism. Does the act of stretching the blocks change the number of blocks? What does this mean about the ratio of the number of blocks in the pyramid to the number of blocks in the prism? How does this affect the ratio of the volumes of the pyramid and prism?
The number of blocks does not change. This means that the ratio between the volume of the prism and the volume of the pyramid is the same as in the previous section. What does this mean for the equation for the volume of the pyramid?
It means that the equations remain the same, but the one that required a cube is now invalid.
Reset all three white dials to 1. What does changing the top and bottom white dials do?
This should appear very similar to what happened when you varied the height of the pyramid. Try using a similar process.
Try varying one of the top white dial or the bottom white dial, while setting the red dial to a high number, and then varying them again while setting the red dial to a low number. What happens?
When the top white dial is changed, the left-right dimension of the pyramid (and each individual layer) changes. When the bottom white dial is changed, the forward-backward dimension of the pyramid (and each individual layer) changes. The choice is arbitrary, but let's call the left-right dimension the length of the pyramid and the forward-backward dimension the width of the pyramid. You could use depth for the forward-backward dimension, but that would require altering the labels in your equations. Furthermore, if you were looking directly down on the pyramid instead then the height would be better to label as the depth. Sticking with the length, width, and height convention (ie. the convention relative to the pyramid) is simpler.
Now, just like in the subsection about the height of the pyramid, how would you deal with the prism that encloses the pyramid?
Just like previously, choose the rectangular prism so that it is as small as possible, while still containing the pyramid (ie. the pyramid fits perfectly inside the prism). Now compare how the length and width of the prism vary. How does this variation compare to the way the pyramid varies? What about the respective volumes?
The length, width, and volume of the prism vary in the exact same manner that the length, width, and volume of the pyramid vary. How do the number of individual blocks within the layers of the pyramid, and the number of imagined individual blocks within the prism compare? Does the ratio between these two numbers change?
The ratio does not change. What does this mean for the equations for the volume of the pyramid?
It means that the equations remain the same.
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?
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?
Digital Manipulative: Volume of a Regular N-gonal Pyramid (and Some That are a Little Stretched)
There are now 5 dials instead of 4. The 4 white dials correspond to the 4 dials in the previous app. The top dial used to be red, now it is white. These four dials behave in the same way as in the previous app. The top one controls the number of layers and the other three control the dimensions. The fifth dial is red. It controls the number of sides of the n-gon. The layers of the pyramid are now made up of n-gonal prisms and there are outlines of rectangular prism layers around them. How do the rectangular prisms compare to the layers of the pyramid in the previous app? What is the same? What is different?
The outline of each layer appears to be the same, however, there are no longer blocks making up each layer, but instead a grid pattern. What do you notice about the grid pattern?
The grid is an equally spaced 3 by 3 pattern on each layer. Does this help emphasize something? Try setting the 4 white dials to 3, 1, 0.5, and 1 respectively. Your figure should have 3 layers that are very short and wide. Drag the layers so that you can see all three of them at once. Set the red (bottom) dial to 8. The n-gon is now an octagon. Does this help?
In each layer compare the octagon to the square face of the square prism outline. Do you see a pattern? Feel free to move the layers to get a good look at each.
Consider the proportion of the area of the square that is covered by the octagon. Does it change between layers? If it does change, how much does it change?
Consider the grid pattern. Does it help you make a conjecture?
The proportion of the areas doesn't change between layers. What does this mean about the proportion of the volume of the octagonal prism as compared to the volume of the square prism surrounding it?
The proportion of the volumes also doesn't change. Are these two proportions (ratios) related?
Yes they are. In fact, they are the same! This is because the volumes are found by multiplying the areas by the heights, the areas are in the same proportion, and the heights are identical. How does this suggest that the volume of the octagonal pyramid compares to the volume of the square pyramid surrounding it?
It suggests that the volume of the octagonal pyramid can be obtained by applying the ratio to the volume of the square pyramid. How would this manifest in the equations for the volume of the octagonal pyramid?
You might correctly assume that applying the ratio to this specific equation for the volume of the square pyramid
is the correct course of action. Why would this be a good assumption to make?
The ratio allows us to convert between the area of the square and the area of the octagon as well as the volumes. When it is applied to the volume equation it can be applied directly to the square base's area, it converts the square's area to the octagon's area. The equation remains the same, but now the base refers to the octagon, not the square. Do any of the other equations convert?
Yes. The equation that relates the volume of the pyramid to the volume of an enclosing prism with the same base also converts.
Why is this?
It is because the volume of the prism is equal to the area of the base times the height. Therefore you can take the equation from hint 11 and replace the area times the height with the volume of the prism. What would happen if a different regular n-gon had been chosen?
The choice of regular n-gon does not matter. An octagon was chosen because polygons with a number of sides divisible by four make it much easier to see the constancy of the ratio of the areas and of the volumes. If a square had been chosen, then there would have been two square pyramids to discuss rather than one square pyramid and one octagonal pyramid. That might have been confusing.
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.
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
Digital Manipulative: A More Direct 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.