How can you find the volume of a pumpkin? How is this related to finding the volume of an ellipsoid? And what does any of this have to do with Archimedes’ famous “Eureka!” moment?
What Is an Ellipsoid?
An ellipsoid is the three-dimensional shape you get if you rotate an ellipse around one of its axes.
To understand how to calculate the volume of these less than perfectly spherical pumpkins, we need to talk about a three-dimensional shape called an ellipsoid. As you might guess from the name, an ellipsoid is the three-dimensional shape you get if you rotate an ellipse around one of its axes (just as a sphere is the three-dimensional shape you get if you rotate a circle around one of its axes). In truth, the shape of an ellipsoid can be a little more complex since it can extend different amounts along each of its three axes, but let's set that extra detail aside for now.
The important point is that an ellipsoid is a pretty good approximation to the true shape of a pumpkin. Some pumpkins are tall and skinny ellipsoids, some are squashed and fat ellipsoids, and some are just slightly ellipsoidal ellipsoids—but they’re all ellipsoids. So if we can figure out how to calculate the volume of an ellipsoid, we’ll be in good shape.
How can we do it?
How to Find the Volume of an Ellipsoid
To see, let’s go back and think about that equation for the volume of a sphere: (4/3)πr3. In particular, let’s think about how we can picture the meaning of this equation. In terms of diameter, the volume of a sphere is equal to (1/6)πD3. Since π is roughly equal to 3 (its actual value is closer to 3.14, but 3 is close enough), this equation says that the volume of a sphere is roughly equal to D3/2. What’s D3? It’s the volume of the smallest cube that contains the sphere. So this equation says that the volume of a sphere is roughly equal to half the volume of the cube surrounding the sphere. Seems about right … right?
The volume of an ellipsoid must therefore be approximately equal to half the volume of its surrounding box.
With this in mind, let’s shift to thinking about an ellipsoid. Imagine taking a sphere and stretching it along its two horizontal axes and squishing it along its vertical axis. What have we done? We’ve created an ellipsoid that is very similar to the classic shape of a pumpkin. Now, what does this stretching and squishing do to the volume? Well, by analogy with the sphere, the volume of this ellipsoid must be approximately half the volume of the rectangular box surrounding it. If we stretch the sphere by 20% in two directions, we also stretch the surrounding box by 20% in each of those directions—which means we increase its volume by 20% for each direction. If we then squish the ellipsoid by 10% in the other direction, we must decrease the newly stretched volume by 10%.
By analogy with how we came up with the relationship between the volume of a sphere and its surrounding cube, if you think about this you'll see that it means that the volume of an ellipsoid must be equal to (4/3)π•a•b•c—where a, b, and c are the ellipsoid's "radii" along each of its three axes (and therefore 2a, 2b, and 2c are the lengths of the sides of the surrounding rectangular box). In the case of a sphere, these three “radii" are all the same, so this equation turns into the simpler V = (4/3)πr3.
How Did Archimedes Find Volume?
Given this new equation for the volume of an ellipsoid, all we have to do to find the volume of a real pumpkin is measure its size (by which I mean its analog of radii) along each of its three axes. Once we do that, a bit of number crunching will give us a new, improved, and much more accurate estimate of the pumpkin's volume.
And while that’s all well and good, and while it very importantly puts us on a path towards better predicting the amount of pumpkin we can extract for pie baking, it’s probably not the easiest way to determine the volume of an irregularly-shaped pumpkin. For that, we really should head back in time a few thousand years and discover exactly what it was that made the famous Greek mathematician Archimedes reportedly run down the street naked shouting “Eureka!”
But, sadly, we’re all out of time for today. So the story of Archimedes and the conclusion to our story of measuring the volume of a pumpkin will have to wait until next time.
In the meantime, for more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Pumpkin image from Shutterstock.