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How to Find the Volume of a Pumpkin

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?

By
Jason Marshall, PhD,
Episode #262

Pumpkin MeasuringPumpkins make for seriously delicious pie, am I right? Right now a bunch of you are thinking “Yep!” and a bunch of you are thinking “Nope,” because pumpkin pie is divisive pie—perhaps the most divisive pie.

Case in point, according to a poll conducted a few years ago by the folks at NPR, pumpkin is right up there with apple and strawberry rhubarb as America’s top pie. A different poll at Epicurious also reported that apple and pumpkin are America’s favorites. But a bit of poking through blog posts and their always fascinating comment sections will quickly tip you off to the fact that for every enthusiastic pumpkin pie lover, there’s an equally enthusiastic hater.

So what’s the math angle to all of this? Why am I talking about pumpkins and pumpkin pie? Well, first of all, it’s fall here in the northern hemisphere which means that pumpkin pie season is upon us. So, obviously, it’s a perfect time to contemplate the mathematical properties of everybody’s favorite squash. Plus, as all of you pumpkin pie lovers will appreciate, the volume of a pumpkin is related to the amount of delicious pie that can be made from it. So pumpkin pie fans need to know the right way to find the volume of a pumpkin! But since pumpkins are often weirdly shaped, finding their volumes can be tricky.

How should you do it? What does it have to do with finding the volume of a shape called an ellipsoid? And what does any of this have to do with Archimedes’ famous “Eureka!” moment? Let’s find out.

How to Find the Volume of a Pumpkin

Imagine you’ve just returned from the pumpkin patch with a big ol’ wagon-full of pumpkins. What’s the first thing you want to do: Carve 'em? Cook 'em? No, obviously you (being the math fan that you are) would want to calculate their volumes. How would you do that? Well, if your pumpkins happen to be perfectly spherical, you’re in luck—because the volume of a sphere is simply equal to (4/3)πr3 (where r is the radius of the spherical pumpkin).

So all you have to do is somehow measure the radius of the pumpkin and then crunch a few numbers. Of course, measuring the radius of a sphere without cutting the sphere in half is actually kind of tough to do. One relatively easy way to do it is to grab a couple of books and position them like football goalposts on opposite sides of the pumpkin. Once you do that, you can measure the distance between the books to come up with a reasonably accurate estimate of the spherical pumpkin’s diameter. With that, all you have to do to find the pumpkin’s radius is divide its diameter in half.

But most pumpkins that I’ve seen aren’t spherical. Sure, they’re kind of roundish, but they definitely don’t look like perfect spheres. What can we do in this case?

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